bargmann, on a hilbert space of analytic functions and an associated integral transform

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL XX, 1-101 (1967)

On a Hilbert Space of Analytic Functions and an Associated Integral Transform

Part 11. A Family of Related Function Spaces Application to Distribution Theory

V. BARGMANN Princeton University

Introduction

0.1. The problems treated in this paper arose in connection with the attempt to apply the methods of Part I1 to the theory of tempered distributions.

Part I dealt with two Hilbert spaces, 8 (= 5,) and 5(= zn), and the unitary mapping A of $J onto 8. (In the following we omit the subscript n as long as the dimension n is fixed.) 43 is the familiar Hilbert space L2( R n ) of square integrable functions y(qj, where q = (ql , . * * , qn) E IR, , based on the inner product

(Yl? YZ)B = 1 Y1(Q)Y2(d d"q , d"q = dq, * * * dq, . 'n

8 may be described as follows. Let 3 (= 3,) be the set of all holomorphic (= entire analytic) functionsf(tj inn complexvariables ( z = (zl,. * * , zJ, z ) = Y ) + <pl j . The elements of 8 are functions f E 3, and the inner product in 5 is

(0.1 )

C , is the n-dimensional complex Euclidean space, (zI2 = zj 1zj12, and dnz = n

dxj d v j .Z In particular, I l f l l Z = (Jf) < 00 i f f € 5. j=1

See [2], hereafter quoted as I or Part I. In the introduction to Part I it was said that Part I1 would deal with harmonic polynomials in 8, and Part I11 with the rotation group. The projected Part 111 has appeared separately ([3]), and some results on harmonic polynomials are included in Section 8 of the present paper.

Unless the domain of integration is explicitly indicated, all integrals extend over the whole range of the integration variables, i.e., Cfl for z and pn for q.

1

2 V. BARGMANN

The mapping f = Ay is defined by

where z2 = 2 z; , q2 = E 45 , z * q = functions xz(q) = A(Z, q) ,

(0.2a)

z j q j , andf E 5 if y E 9. In terms of the

f ( z ) = ( x z 9 Y)s, = (Y , X J 5 j 7 f = A y .

I t will be convenient to define, for f E 3, (0.3) f + ( Z ) = eZ”y-(z) , f-(z) = e-Z”;”f(z) ,

0.2. A tempered distribution (or “distribution”, for short, since no others will be considered) i s a continuous linear functional, say I ) , on Schwartz’ space Y of test functions, i.e., a linear mapping

Y - {v, Y > 9 Y E 9 2

of Y into complex numbers. A complex valued function y(q) belongs to 9 if it has continuous partial derivatives of all orders which decrease fast at infinity, so that for all I E N and all multi-indices m = (ml , * , m J , m, E N,

here N denotes the set of non-negative integers. The topology of 9’ is that of a countably normed space with the non-decrtasing sequence of norms c, = max c ~ , ~ , k = 0, 1, 2, - * .

The distribution v is regular if, for a suitable function u ( q ) , L+(m\=k

(u , Y> ‘S .ov(4) d“q *

If u ( q ) E &, then {u, Y > = ( v , Y)E, = (g,f ) g , where g = AD, f = Aw; g(z) = {u, xz) is holomorphic (see (O.Za)), and

0.3. The treatment of distributions to be discussed in this paper is based on the crucial fact that (0.4) is valid for all distributions ((71: xz> being always defined

HILBERT SPACE OF ANALYTIC FUNCTIONS-I1 3

since xz E 9’). More precisely: Set, for g, f E 3,

(0.5)

- whenever the integral converges absolutely. Then 1) g ( z ) = {v, xz} is holomorphic in z for every distribution u, and 2) {u, y } = (g,,f), wheref = Ay. (This is proved in Section 2.8.)

0.4. In addition it is desirable to characterize the functions f and g which occur here more directly. Since Y is evidently contained in 8, A maps 9’ onto some linear subset @ of 5, and the topology introduced in Y imposes a certain topology on ‘3. Correspondingly, 9’ (the set of tempered distributions) is mapped onto G’ (the set of continuous linear functionals on E). Since these functionals have the form (g,f)-with a uniquely determined g-the set @’ may be identified with the set of all g.

It turns out that the function spaces @ and E‘ may be easily defined by the growth properties of their elements. Set, for all real p,

l f l , = SUP e , w If(4l 2 f E 3 , Z

where @(z) = ( 1 + (z12)P/2ealz12/2.

norm / . I p . If u > p , then Iflo 2 Ifl, and hence @“ c EP.

Let GQ be the set of allfE 3 for which If[, < 00. GP is a Banach space with

The definitions of (5 and (5’ are now simply

m m

e = n e k , E’ = u e - k , k E N , k=O k=O

or equivalently, since CP+l c @p,

+oo +m

E = n ~ k , el= u ~ k , k=- m k=- w

so that E appears imbedded in G’. The topology of E (induced by the mapping A Y = E) is again that of a countably normed space-with norms ).Ik.

The mapping u ---f g, u E Y’, g E (5’) will also be denoted by the symbol A. Thus

(0.6) (Ay)(z) = {y, x Z > J = {’, x%> 3

and {u, w} = (Av, Ay).

0.5. Instead of the spaces E, one may use a closely related family of Hilbert spaces 3 p , - co < p < co, to define @. 5 p consists of functions f E 3, and f

The inner product of two elements f, g of 5” is ( g , f ) , = J g ( Z ) f ( z ) d,uP,(z). Again 5” c s p if c > p, and 5O = 8. Furthermore, it may be shown that E p f n + 1 c 3 p c Ep. It follows that

i- co +do

E = n s k , E’= u g k . k=- m k=--m

As was proved in Part I, Section 1, the monomials

u, = W/d&Zj ) 2[”1 = z? ) [m!] = IT m j ! , I 3

where m is any multi-index, form an orthonormal basis of 5. I f f (z ) = zm c(,u,(z) belongs to some g p , then a, = (u, , f) and, for f E e, g E E’, one obtains the absolutely converging series

( g y f ) = 2 (9, um)(Um ,f) - m

- s p , in turn, has the orthonormal basis ufn = u m / z / q f m I , where the constants

7; (s E N) satisfy the inequalities ci(n + s ) P 5 7; 5 cz(n + s)P for some positive c : , 6 ; . Thus

Lastly we recall from Part I that

A9, = u r n , q m ( q ) = n [ 2 m , m J ! l / ; ; ] - w m j ( q J e - Q : / 2 ; 3

H,,, (4,) are Hermite poIynomials.

0.6. The results about distributions mentioned so far may now be restated as follows.

1) A bounded continuous function y(q) belongs to Y if and only if, for every k E N, zpa ( n + Iml)k ]{pm , y)12 < co. Conversely, if a set of coefficients a , is so chosen that C (n + I K , ~ ~ < cc for all k E N, then y = 2 a,~), E 9.

2) If u E Y’, then {u, y> = 2 , {u, ym>{plm, y } and, for some k ,

2 (. + Iml)-k I{% 9)m>I2 < 02 . m

HILBERT SPACE OF ANALYTIC FUNCTIONS-11 5

If a set of coeficients Pm satisfies the inequality C (n + I o L ~ ) - ~ / B n C l 2 < co for some k , then {v , y } = z,?& bm(pm , y } is a distribution, and (21, p,} = bm .

‘This characterization of Y and of Y‘ in terms of the “Fourier coefficients”

znL = { F , ~ , y} and = {v, T,,~} appears already in Schwartz’ treatise [ l l ] (Volume

Schwartz introduces, however, only the coefficients u r n , f i m but not the holo-

-

11, pp. 117-118).

morphic functions f = I; u,u, and g = C Pmum constructed from them.

0.7. I turn now to a brief description of the content of this paper. There are three main subjects. (a) The mapping f = A y and its inverse (Section 1). (b) The

function spaces Ep, 50, E, E‘ and their interrelations (Section 2. The spaces @‘, @, and E’. Section 3. The spaces S P . Section 4. Convergence in (5’. Linear operations on (E and E’.) (c) Distributions (Section 5. Structure theorems and basic operations. Section 6. Test functions and distributions of compact support. Section 7. Periodic distributions and their Fourier expansion. Section 8. Homo- geneous distributions.) In addition there is an appendix (A) on estimates and inequalities for the functions 0; and for Hermite polynomials.

(a) In Section 1 the mapping A is not restricted to the elements of Y but is applied to a wider class of functions. The dependence of Ay on the growth properties and the smoothness of y is examined.

(b) The spaces (EP, S P , etc. are studied for their own sake, i.e., in greater detail than is required for the purposes of distribution theory. As is usual for Hilbert spaces of analytic functions, every 5 p has a “kernel function” or “reproducing kernel”.3 It is also shown that the family 5” provides a simple explicit example for a Gel’fand triple (“rigged” Hilbert space).

ic) The treatment of distributions calls for more extensive comments.

0.8. The introduction of the function spaces (3 and (E’ provides an auxiliary tool for distribution theory. The value of such a tool may be judged, in part, by the ease with which certain basic operations (defined on Y or 9‘) may be carried out on the elements of (f and (5’ or with which certain classes of distributions (distributions of compact support, periodic distributions, etc.) may be defined in Q?. In some cases-but by no means in all important cases-the situation is indeed quite favorable.

For example, let v E Y, and set v1 = a v / a q i , v2 = q p , u3 = Fv (Fourier transform), where the vk are constructed according to the definitions of distribution theory. If g = Av, and g, = Avk, one obtains g;(z) = 2-W3-/azi, g;(zj = 2-l”9g’/azj (see (0.3)), and g3(z) = g( - i z ) . On the other hand, convolution does not seem to lead to a simple operation on E’.

Once the mapping A of Y, Y‘ onto E, (5’ has been established, i t is found, in addition, that a number of general results about distributions may be easily

See Bergman [4] and hron3zajn [l].

6 V. BARGMANN

inferred by standard analytical methods, among them Schwartz’ theorem that every tempered distribution is the derivative of a continuous function of polynomial growth, and also the “nuclear theorem”.

0.9. In order to explore these connections in a systematic fashion I chose the following procedure. Only the definitions of 9’ and 9” but no further results are assumed from the start, and the theory is mainly developed in terms of the function spaces CZ, E’. The main facts about Hilbert space are taken for granted, but otherwise the paper may be read without detailed knowledge of the modern theory of topological vector spaces.

It is well known that the theory of tempered distributions has obvious applications to the distribution theory on Schwartz’ space 9 because 2 c 9’. (Every tempered distribution belongs to 9, every distribution on 9 with compact support may be interpreted as a tempered distribution, etc.) But no further attempt has been made to answer the question whether the mapping A or a suitable variant may be fruitfully applied to the distributions on function spaces other than 9’.

The discussion of distributions in this paper is of course based on Schwartz’ classical treatise. But I should like to record also my great debt to the work of Gel’fand-Shilov-Vilenkin [6 ] , [7] on “generalized functions”. Many of their ideas and of their methods are used here.

0.10. Formulae, theorems and propositions are consecutively numbered throughout a section. Lemmas and definitions are consecutively numbered throughout a subsection. A list of symbols and of definitions is included at the end of the paper.

1. The Mapping A and Some Function Spaces Related to it la. The Function Spaces 9: and Yk

1.1. Preliminary remarks and definitions. The real and complex number fields are denoted by F? and @; R, stands for the set of non-negative real numbers, and N for the set of non-negative integers. The points (or vectors) of R, and C n (real and complex n-dimensional Euclidean space) are usually denoted by q = ( q l , * * , q,) and z = (zl , . 1 . , zn) , respectively. We write

a - 6 = 2 ajbi for the inner product of two vectors a, b and la] for the norm

(Ij ( a j ( 2 ) 1 / 2 of a (both in the real and the complex case); a2 stands for a * a =

n

i=l

Frequently a vector z in C n is written z = x + 6, x,y E IR,, so that, for example, 2 2 = x2 - y 2 + 2 i x -JJ and [ zI2 = x 2 + y2.

1.2. Standard decomposition. Let t = x + iy E C, . Then we may set x = t a , where 6 = 1x1 5 0, a is a unit vector in R n (i.e., la1 = 1), andy = qa + b,


b E Ip., , b * a = 0. Thus = 5a + i b , 5 = E + iq E C+ ( 5 2 0 ) ,

a, b E R, ; la1 = 1 , b - a = 0 , (1.1)

(2, denotes the half-plane 3-?e 5 2 0. By means of this standard decomposition the analysis of a function f ( z ) on

C 71 may often be reduced to the analysis of a function y ( 5 ) = , f ( (a + b ) on C+ . Yote that 1 + [zI2 = l2 + 1 = d m , and

(1.2) a 11 + tl 2 < = 1% + 1p = 1 + (z(Z 2 Il + 5 1 2 .

1.3. Monomials and derivatives, A multi-index m is an ordered set of n non-negative integers (m, , * . - , mn), and m = 0 if all mi = 0. Sum and difference, mb = m f m‘, have the usual meaning (m; = mi f mj) , and we write m m’

(or m‘ 2 - m) if m j 5 mj for all j . We set Im[ = 2 mi and [ m ! ] = IT mi! . n n

5=1 j=1

The monomials qIml and zLml are defined as IT q? and zim, . The operators j i

a/aqi and a/ati are denoted, respectively, by a j and d j . For any m,

Occasionally it is useful to introduce

(1.3a) irm1? = a[m]q)/[m!] , %rmlf= d[”lf l[m!] , so that the Leibniz rule takes the simple form

(1.3b)

1.4. The functions 6; . For a convenient definition of the growth properties of various function classes we introduce, for real a, p ,

(1.4) e;(s) = P’’Z( 1 + s 2 ) p / 2 , S E R , .

For any u in IR, or C m we define

(1.4a) e p ) = e;(lul) = e a l u l * / y ~ + l ~ 1 2 ) ” / ~ I

In particular, e p ) = ( I + ( u 1 2 ) ~ ’ 2 , e; = eajuiP/2. Clearly,

(1.4b)

(1.4c)

0;. q; = ,ga+a’ P+P’ ’ e; e1; = I ,

e; (u ) 2 e;:(u) if a S a’, p 5 p’ .

8 V. BARGMANN

These functions will be frequently used in most parts of this paper. More detailed information about them is collected in the appendix (see AII).

1.5. The classes 9; . As usual we denote by Co the class of complex valued continuous functions y(q) ( q E IR J, by Ck ( k E N) the class of complex valued y(q) with continuous partial derivatives up to (and including) order k, and by C” the class of functions with continuous partial derivatives of all orders (i.e.,

0)

ca = n c k ) . k=O

9; consists of all functions y E CO which satisfy an inequality of the form

I Y ( d 1 s @;(q) = Y ( 1 + q2)p’2 for all q

with a constant4 y depending on y (functions of-at most-polynomial growth). If p < 0 these functions decrease at infinity.

DEFINITION OF 9; . Afunction y(q) belongs to 9; if (i) y E Ck, (ii) there exists a positive constant yo (depending on y ) such that

(1.5) lacmlY(q)l 5 Y o q d if 14 = k *

The following simple result will be useful.

LEMMA 1. k

For p 2 0, 9; c n L Y ~ + ~ - ~ , i.e., (1.5) implies the additional l=O inequalities

(1.5a) larm’y(q)l 2 y V 6 : + v ( q ) ;f I m l = - J

v = 0, 1, * * * , k , with suitable positive y, . Proof by induction on Y : Let v 5 k, suppose that (1.5a) holds for v - 1, and

let P be a bound for all lacmly(0)(, Iml = k - v. Now

f l n

Since c lqj ( 5 4; 1q1 < l/n O!(q), and O:+v-l(q) 2 6z+v-1(q), we obtain

l a c m l Y ( ~ ) l s P + Yv-l& 1qA 6;+”-l(d 5 l’v6;+v(q>

with yv = P + l/; q.e.d.

1.6. The spaces Y k and Y. L. Schwartz’ space Y may be defined as follows: a function y belongs to 9 if it belongs to all 9; .

The terms “constant” and “positive” always refer to a finite number.


It will be advantageous, however, to consider first a certain family of function spaces Yk, k E N, and to introduce the following definitions.

DEFINITION 1. For any y E Ck, the norm (1 y 11; is given by

m=, {SUP lft.-,,l (4) acm’Y(!7)l> * ImlSk QER,

IIY 11; =

DEFINITION 2 . Yk is the set of all functions y E Ck for whic:!z IIy 11; < co. “

DEFINITION 3. Y = n yk, i.e., Y consists of those functions y E C“ for which all norm 11 y 11; are jni te . ’=O

Yk is clearly a normed linear space with norm 11. 11;. Since O:(q) s €$,(q) if

If y E Yk, then la[mly(q)l ye!!k+,,I(q), y = Ily [I;, ~rnl 2 k, and it foIlows

easily that Yk = n Y d ( k - 1 ) .

p < p’, it follows that Yk+l c Yk, and that IIy 11; s Ilyl$+, for any y E Yk+l.

k

z=o Lastly, the topology of Y is defined as that of a countably normed linear space

(see [6], Volume 11, Chapter I, 93) i.e., the balls IlyII; < (1 + 1)-l for all non- negative integers k, 1 form a basis for the neighborhoods of 0.

1.7. The class 3. The class of all holomorphic (entire analytic) functions f( z) on C , is denoted by 3 (or 3 ,) . For any f E 3 we define

(1.6) f + ( z ) = ezz//”f(z) , f - ( z ) = e - ” / ~ ( z ) , so that

(1.6a) If+(z) 1 = e(2*-gz)/2 ~ f ( z ) I , ~f-(z) I = e ( V z - z s ) / 2 ~ f ( z ) I . An element f of 3 may be expressed as f (2 ) = 2, amz[ml. More frequently,

however, we shall write

(1.6b)

The functions u, form an orthonormal basis of the Hilbert space 5 (= 5,) (see Part I, equation (1.6)).

lb. The Mapping f = Ay

1.8. The mapping f = Ay (or f = Any) is defined for measurable y(q) if the integral

f (4 = /&> d Y ( d d”q ,

A ( z , q) = T - ~ / ~ exp { -i(za + 42) + 2 1 / 2 2 - q} ,

dnq = dq, * * - dq, .r (1.7)

10 V. BARGMA"

is absolutely convergent for every z E C n . In the present section only continuous y will be considered.

In accordance with (1.6) we also introduce A*(z, q) = e * z * / 2 A ( z , q) and the corresponding mappings f+ = A'y, f- = A-y, where

n "

(1.9c) d["lA+(Z, q) = 2lml/2q[m'A+(z, q) . Equations (1.9b) and (1.9~) are closely connected with the relations on p. 189 of Part I which led to the construction of A.

The power series for A reads

(1.9d)

where plm(q) = (21"1[m!]~~/2)-~~~H,(q)e-~~/~ are the normalized Hermite functions (see AIIIa and Part I, equation (2.10b)).

One finds €or the moduli of A- and A

Let f = Ay. The analyticity off (z) is established if any one of the integrals in (1.7) or (1.8) is analytic in z. Thusfis holomorphic ( f ~ 3) if, for example,

(1.11) J

because IA+(z, q)l 5 T - ~ / ~ ~ - P * / ~ P " J I for IzI I - u / f i (see I, p. 191, Theorem C). Forf+ = A'y we obtain then, from (1.9c),


and on inserting the expansion (1.9d) in the integral (1.7) we find

(1.Ilb)

The condition (1.1 1) is amply satisfied if y is a function of polynomial growth. To obtain an explicit estimate let

(1.11 c) IY(d1 s Y q d .

( I . l ld ) If-(z) 1 =< yKIP '7 fn '4 - l ip[,n 'l q ( x d 2 )eY2 (by (A. 10a)),

Then, by (1.lOa)) If-(z)l j y ~ r - ~ / ~ e " * J O;(q)f&'(q - xd2) dnq, hence

(I . l le) ~f-(z)l j y(4Tr)n/4ev2 if p = 0 .

LVe conclude this subsection with the following uniqueness proposition.

LEMMA 1. Ifyl , y2 are continuow functions of polynomial growth and Ayl = Ay2 ,

Let y = y1 - y 2 . Then f = Ay = 0, so that f t ( z ) = ez ' / " f (z) = 0 for all z, in particular, f+(;r ) = 0. Apart from a constant factor,f+(+) is the Fourier transform of the square integrable function e-**/2y(q) (see (1.8a)). Hence

then Y I ( 4 ) = y2(4)for all q.

Proof:

Y(9 ) = Y l ( d - w2(4) = 0 for all 4.

1.9. Functions in 9': , k > 0, p 2 0. If y E 9'; , the estimates (1.1 Id), (1.1 1 e) may be refined. This is done in Sections 1.9-1.12.

If y E 9'; , then y has continuous partial derivatives of polynomial growth of all orders 1 j k (Lemma 1, Section 1.5). In (1.8) thederivatives d["] f - ( z ) may be taken under the sign of integration. Using (1.9b) we find, for In( j k,

12 V. BARGMANN

1.10. Evaluation of f-(2). Set, for fixed z and real T , F ( T ) = ~ - ( T z ) , so that F( 1) =f-(2). Then F ( z ) ( ~ ) = dzF/dd is given by

Ifqz(z) is a bound for all Id["]f-(z)l with Iml = 1, we find

(1.13a) r z ( 4 5 (dn 14)z r z ( T 4 *

By Taylor's formula,f-(z) = F(1) = p , ( z ) + rk ( z ) ,

( 1 .13c)

ESTIMATE OF l rk (z )] , Suppose that y satisfies the inequality (1.5), i.e., Combining (1.12) and (1.lld) one obtains larmly(q)l 5 yoO:(q) if Irnl = k.

therefore a bound qk( z ) of the form

(1.14) qk(z ) = 2k/2/30;(xd2)eya , /3 = yoKPTni~~,, *

Since O : ( x d 2 ) 5 2P/20:(x), we find from (1.13a)

(1.14a) J F ( ' ) ( T ) ~ 2~ '2 j3~k0~(~)eT2y2 , 5 = d2n 1.21 . Note that erz'vz 5 ery2 5 - e r ( l i V a ) , 0 2 T 5 1. Hence Irk(z)l is majorized by

Set E = 1 - T . Then, for positive u,

Hence w,(l + y 2 ) 2 eO?,,(y), and we find the desired estimate


1.11. We may now prove the following result.

THEOREM 1.1. r f y E P:, k > 0, p 2 0, then f = Ay is holomorphic, and

If(')/ 5 y ' e t + k ( x ) e y k ( y ) with some constant y'.

Proof 1) Since < = d/2n 121 and l z l p < O:(z), (1.15b) implies that Ir,(r)l 2

2) By (I.l3b), lp,(z)l 5 B20i(z) for a suitable B 2 . Now O,O(x) 2 1, and

ple;(x)O!zk(~)O;(z) for some p1 .

r t ke ! z , k (y ) 2 (by A*8)> thus I p,(z)l 5 B2T~,e,0(x)el,,(y)e,o(z).

3) Lastly, O!(t) 5 O ~ ( x ) O ~ ( y ) (by (A.6a)). Hence

If-(')l Ipk(')l + Irk(')! 5 y'6:+k(x)e!k(y) 3

with y' = B1 + /127ik . Since If(.)! = e("*-y2'/2 lf-(z)l, the theorem follows.

Remark. A direct application of (1.1 Id) would only yield the weaker estimate \ f ( z ) l 5 y'O:+k(x)elZ1*/2.

1.12. Functions in Yk. For these we obtain a considerably sharper result.

THEOREM 1.2. I f y E Yk, then f = Ay is hofomorphic, and

I f ( ' ) l 2 ~ ~ ~ ~ ~ ~ e ~ k ( z )

with a constant ak independent of y.

Proof:

2) If k 2 1 , we conclude from the definition of /Iy 11; in Section 1.6 that

1) If k = 0, the assertion follows from (1.1 le), and a, = (4~) "'4.

(1.16a) lacm'Y(d l 5 11Y 11; 9 I m l 5 J

( 1 . I 6b) IY (d1 5 l IY l l ~~ " (4 ) *

I t follows from (1.16a) that a relation of the form (1.14) holds for 1 5 k , with p = 0 and yo = I iy I / f , i.e.,

q J z ) = 2 1 ' 2 / 9 e ~ 2 , 0 g I 5 k , p = (4,)n'4 i\yllf

so that, by (1.13b) and (l.l5b),

14 V. BARGMANN

3) Let 6, = min (1, (elk)'"), i.e.,

(1.17) 6, = 1 if k 2, 8, = (elk), if k 2 3 .

Since (by (A.8)) eOEZk(y) 2 (elk), 2 8, , one finds k

If-(')l 2 k k ( ' ) l + Irk(')l p ( e / 8 k ) 0 2 2 k ( Y ) I: 5' 1=0

5 @(3e/26,) (2n)k/2eg(2)e?2k(Y) ,5

i.e.,

(1.18a) If-(z)l 5 p' IIylI~8~(z)O0,,(y)e"' , p' = ( 4 ~ ) ~ ' ~ ( 2 n ) ~ / ~ ( 3 e / 2 6 , ) . 4) Using (1.16b) one obtains from ( 1.1 1 d) a second estimate :

= KLnn/4 '1 ( 1 . 1 8b) If-(z)I s P" l l Y 1 / ; @ ! k ( x ~ ~ ) ~ y 2 Y Jk,n '

A comparison of (1.18a) and (1.18b) shows that

( 1 . 1 8c)

In fact, if Iyl 2 1x1, theny2 2 4 1zI2, O!!zk(y) assertionfollowsfrom (1.18a). If lyl < 1x1, then IzI < 1x1 2 / 2 , O ! , ( x 2 / 2 ) and the assertion follows from (1.18b).

k 2 1. Thus p = 2,p', i.e.,

(1.18d)

If-(') I 5 f IlY l/;e%(2)ey' J p = max (2,p', p") . 2k0802,(z) (see (A.7)), and the

O!,(z),

By a closer analysis-based on (A.13), (A.13a)-one finds that 2"' 2 p" if

p = uk = $ ( e / S , ) ( 4 ~ ) ~ / ~ ( 8 n ) ~ / ~ .

This establishes the theorem because I f ( z ) l = e("a-u2)/2 If - (4 I f

fi(z) = (4n)n/4ez2'2,

1.13. Some examples. For later use we list a few pairs y , f ( = A y ) .

1 . y l ( q ) = 1 , f - ( z ) = (4n)n/4,

(4n)n14 A

2. Yn(4) = qCml , f d z ) = 21""2 (4V)n14 i i m ( z ) e Z 2 / 2 , -f ;(z) = 21"112 H , ( z ) ,

3. ys(q) = t b ( q ) = e ib .a ,

4- Y d Q ) = XJ4) = 4 4 P) , 5. W 5 ( d = v m ( d 5

f3(z) = ( 2 ~ ) ~ / z A ( i z , b ) , b E R n ,

U E C n , f 4 ( 4 = e , (4 = 8"", f5(4 = u * ( 4 *


Remarks. Examples 1 and 3 are obtained by direct computation. One obtains fi fromfi by applying (1.lla) tof:(z) = (477)"'*ez2 (see AIIIb for the definition of i?,,,), To obtain f4 use

A ( d , q )A(b , q) d"q = ed'b s (1.19)

(see I, equation (2.2), p. 198). To obtain f5 use (1.9d) and the orthonormality of the functions P),,, .

lc. The Function Spaces D; , @' and (-%

1.14. The spaces Q; . The two theorems in Sections 1.1 1 and I . 12 suggest the introduction of linear function spaces in 3 which correspond to some extent- via the mapping A-to the spaces 9; and Yk. For the sake of greater generality, however, the index analogous to k will no longer be restricted to non-negative integers.

DEFINITION OF Q; . A function f E 3 belongs to Q! if, for some positive constant y ,

~f(z)l 5 ye;(x)e:p(y) = ye",x)e"(y)elz1a'2for all z E c,. Here, p and p may be any real numbers.

Note that Q$ c Q$ if p 2 p', p 2 p'. Theorem 1.1 may now be more concisely formulated. We combine it with the

inequality (1.1 Id) and obtain

THEOREM 1. la. AS: c Q:,., if p 2 0, AP: c D: for all p.

1.15. The spaces (5,. We first introduce the following norms.

DEFINITION 1. Iff E 3, then for evey real p the norm If I p is given by

If lp = S U P q'(4 I f (4I '

If I p = SUPz e-Xa6;(4 If'(4l = SUPz e - q ( z ) If-(4l Y

z e c *

Equivalent definitions are:

(1.20)

( M ( Y , ~ ) is a non-decreasing function of r. For positive r, M ( r , f) > 0 unless f = 0.)

The norm Ifl, is defined for all f E 3, but it may assume the value + 00.

DEFINITION 2. Q? consists of all functions f €3 f o r which If[, < co. (Ep is a

normed linear space with norm ) . I p .)

16 V. BARGMANN

DEFINITION 3. C? is the intersection of all Ek, k E N, Cn

0; = n ~ k . B=O

Cf is equipped with the topology of a countably normed space, i.e., the balls l f l k < ( I + 1)-l ( k , I E N ) form a basis for the neighborhoods of 0. ( I t suffices to choose the balls ] f l k < (k + l)-l as such a basis (see (a), Section 1.16).)

Theorem 1.2 may be stated as follows.

THEOREM 1.2a. (i) A Y k c Ek, and hence A Y c (5.

(ii) r f y E Y', then lAylk 5 a, Ilyil:, with a constant a, independent ofy.

1.16. We draw a few simple conclusions from the definitions, postponing a more detailed analysis of the spaces (5, to Section 2.

(a) I f f € W, then, for all z , If(z)l S Ifl,@,(z), If+(z)l 5 Ifl,eZ2O!!, , and If-(z)l 5 Ifl,eyzP,(z). Conversely, iff E 3 and if, for all z E Cfl , I f ( z ) I 5 ce:,(z) (equivalently, ~ f + ( z ) l 5 cer*e!,(z) or ~f-(z)~ 5 ceU%:,(z)) for some positive constant c, then f E (5, and If/, 5 c.

(b) If p < 0, then O;l(z) 2 O;'(z) so that I f l , 5 / f l u for everyfE 3. Hence

p < G + . (50 c C?,,

Therefore, 6 may also be defined as 'w

e = n E,. k=-m

In particular, (5 c (5, for every p.

sequencef, E (f such that Ifkl,,/lfklp + co. (c) If cr > p, the norms I.],, and 1.1, are inequivalent on (5, i.e., there exists a

Proof: Set f k ( z ) = (1 + z ~ ) ~ , k E N. Then M(r, fk ) = (1 + r2),, = @,(r ) so that (by (1.20a) and (A.8)) IfJ, = ~ t + ~ ~ and I f k ! , = ~ i + ~ ~ . Now T$+?,/T; t-2k - (2k)"-P''2 as k + co, and the assertion follows.6

(d) I f f # 0, then limp-+m If], = co.

(Proof Letf(a) = c # 0, a # 0. Clearly Ifl, 2 1c1 O;l(a), and O;'(a) -j a3 as p + + co.)'

Here, and later, Pk N ar means lim, P k / x k = 1. ' On the basis of (1.20a) one may state more explicit conditions for the growth of Ifl, . Let

p 2 2, thcn (by (A.8)) O;'(rJ = T: for Y, = d p - 1 2 1, hence I f l , 2 B; l ( rp )M(rp ,f) 2 ~;.W(l,.f) . Thus i f f # 0, then Ifl, 2 YT; (for all p 2 0) with some positive y. (For f = 1,

-

If I p = 7; .)


(e;! I f f belongs to theHilbert space 5 studiedin Part I, then I f ( z ) ( 5 / / f l l e i z / 2 / 2 (see I , ;1.7j, p. 193). It follows that 5 c and lf lo 5 ilfll.

( f l lidations between Qi and 0.". A simple analysis shows that the following relations hold: (i) If one of the two numbers p, - p is non-negative, then E; c P ( G = min (p, -p)j . (ii) If p and -p are both negative, c 0 . p - Q .

(iii) If cr 2 0, 0." c 0:: ( p a n d -p 2 0, p - p = a). (ivj Ifa < 0, 6" c QEu.

1.17. The classes 6, and the operators 8, . In Part I (Section ig, p. 197) the classes Gj, were introduced for ?, < 1. We first extend this definition to all po.rilir'u I.. Thusf(E 3) belongs to GA if, for ail z , If(zj1 5 ye'ziz'2'2 = yBt(l.zj for some positive constant y . Evidently 8, = 0.O.

For any positive k t h e operator 8, is defined by

( Q A f ) ( 4 = f ( W . Clearly Q,Q,, = 8,,, , and Ql = 1 (identity). Also QAQ,. = BAx .

spaces 6, and EP and on the action of the operators R, on them. The following lemmas contain some results on the connection between the

LEMMA 1. In,f/, Ifl, i f 2 < 1.

Proof: z\f(r, Q,f) = M ( I , r , f ) 5 M ( r , f ) , and the assertion follows from (1.20a).

LEMMA 2. I f 2 < 1, then,for anypair a, G', I8,fIu 5 cAaa, Ifl,. with a constant c ~ , ~ , . independent of$

Proof: The aqsertion is implied by the inequality

(1.21) 6;'(2) 5 l-1u1T;-u,0;1(3,z) (0 < 3, < 1 , o! = - 1 )

which may be established as follows. Set w = lz, and = 1 + a. Since O:(klw) 5 k l u l O ~ ( w ) , we obtain

e;l(?.-iw) = 0 g ( k 1 ~ ) 0 ; - l ( 4 5 A-l+;yw)e;yw) _< - n-l"17~-,.eg.(~)e;l(w) = A-'"~;- , .o;?(w) ,

using (A.9).

the assertion follows from the definition of the norms / . I p . By (1.21), 6;'(z) I f (Az) l c,.,,,O;?(ilz) If(3,z)I, with c~,~,,, = klul~;-u,, and

COROLLARY. r f J . < 1, then sZ,O.'' c @a f o r any G, a'. (Equivalently, Q,@Q c 0. for all p.j

LEMMA 3. For all real p, (i) 0.P c 6, if p > 1, (iij 8A c 0.' ;f 1 < 1, (iiij Q,EP c 8,. if?, < it.

18 V. BARGMANN

Proof: The only properties of the spaces (I? used in the proof are those stated

2x1 .

(Therefore, the lemma also applies to the spaces 3 p to be introduced in Section 3.) (i) Let f E CF. Then g = R,-1 f E (Eo c Gl, hence f = Q p g E 8, . (ii) Let

f € O n . Choose A' such that il < A' < 1, and set x / A = v > 1. Then g = R, f E (iii) Let f E (50, and set q = A/A' < 1. Then g = Q, f E go c 01, hence QA f = QArg E Q X .

in the corollary and the inclusion property

@A = (50 = 0 1 ,

c CEO, hence f = Qv-lg E Ep.

1.18. The form (g,f). For two functionsf, g E 3 we define

(1.22)

whenever the integral converges absolutely. The bracket (9, f ) coincides with the inner product (g, f) iff and g belong to

the Hilbert space 8, but it preserves the properties of an Hermitian bilinear form for a wider class of functions. ((5 g) = (9, f ); (9, Pf) = P ( g , f ) for any complex

It will be shown in Section 2 that every continuous linear functional L( f ) on &--and hence every tempered distribution-has the form L( f ) = (g, f) with a uniquely determined g. Thus the form ( , ) is particularly important for the subject of this paper.

-

Constant B ; (g , f l +fz> = (g,f,) + (g, f 2 ) if both ( g , f j ) are defined.)

LEMMA 1. Let g E EU, f E @, and p + c > 2n. Then (9, f) is defined, and

(1.22a)

Proof: The assertion follows from (A.17a). Since If(z)l 5 If IpOlp (z ) and 1g1u05u(z), the integral (1.22) converges absolutely and is majorized by lg(z)I

'0 ~ - p - - a , 2 n 191, I f l p .

- 'THEOREM 1.3. Let I?,(,?) = e'", U n a ( Z ) = 2rm1/z/[772!]. If

f tz) = z: ~ ? n U , ( Z ) E @ A m

and A2 < 2, then

(1.23) (e, , f ) = f (a) > (urn 3 . f ) = X m *

Proof: 1) Choose a positive v such that 12 < v2 < 2, let 7 be any complex number of modulus 171 < v/L, and s e t f , ( z ) = f (72) = Em ~ l " l c (murn (z ) . Then,


writing h for either e,! or u, , we find that

1(7) = (h,f,) > 171 < 4). , i5 analytic in T. y ~ h ( ~ ) ~ e ~ * ~ ~ ’ * ’ ~ for all 7, and the integral 4 112 z j e Y 2 z ’ 2 2 ) I f f belongs to 5, equation (1.23) is valid (see 1 , bection 1) . 3) If/?-) < v-l , thenf, E 6Jv-~l c 5, hence (e,, ,f,) =J;-(a) = f , ~ a ) ,

and u,,~ . f-‘ = ~ [ ‘ ~ l r , , ~ . Equation (1.23) follows by analyticity.

In fact, l h ( z ~ ~ \ z j l d p I L ( z ) converges

(;OROLL,ARY. I f f E A2 < 2, then

(1.23a)

Proof: Ry Theorem C in I, Section 1,

ah’aa, = J ( a e e j / a u j ) / ( z ) dpn(z) .

‘ h e theorem applies to the elements of any space ‘3’’ (see Lemma 3, Section 1.17; or 2; . It also applies to the elements of the spaces j-Jp to be defined in Section 3 . In all these cases

i 1.23h i f = 2 amurn = 2 (urn ,f >urn m m

Id. The Mapping cp = Wj

1.19. It was shown in Part I that A-* is related to the operator W which may he defined a3 follows. I f fE 3,

q~(qj = (wfj(q) =j .4 i i , q ) f ( z ) dpc,(z) =p-(~, q)eiZ/y(z) d p , ~

whenever the integral converges absolutely. ‘Yo obtain criteria for the continuity and the differentiality of pl set

B [ ? ~ ] ( z , q ) = l i Y m l ~ - ( i , q ) e i * ~ ~ ( z ) e - ~ ~ ~ ’ ~ .

If, for every m of norm Irnl bounded sets in R then E Ck and (for IrnJ k )

k , S B [ m ] ( ~ , q ) dnz converges uniformly in q on

20 V. BARGMANN

THEOREM 1.4. Let f E 3, and 1 f (z)I 5 @ p ) O $ ( y), where ,u = k + n + 7, k EN, T > 0. Then p = Wf E Ck, and

Iacmlp(q)I S ~ ' p , ~ , ~ m l f $ ( q ) > Iml 5 k >

with constants b independent off. Hence WQ; c 9; for all 1 5 k.

Proof: We start from (1.26). Since e j (q ' ) 5 2 l P l / W ~ ( q ) , it follows froin (A. 10) and (A. 10a) that the integral in (1.24b) is majorized by the expression

y(n- n/42( I" I+lUl)/2 IP I PI WL 1 $If I m I, n jpm l-op,nPpO (9 ) , which defines the constant bp,p,lml .

we state here the following. Several variants of this result may also be of interest. For a later application

71

THEOREM 1.4.a. Let f €3, and If(.)[ yn[O~(xj )O.! , (y j ) , where ,u > 1. j=1


Then y = Wf E Co, and Iddl 5 Y(b;,ll)” rI O,o(q,) .

i

Proof: Since d,u,(z) = dpu,(z,) and A ( z , qj = A,(t , , qi ) , the problem J 3

is reduced to n one-dimensional ones of the type just treated. In addition, no derivatives occur so that = 1. Thus we conclude from Theorem 1.4 that

1.20. ‘The most important result in this context is

THEOREM 1.5. If p = n + k + 7, k EN, 7 > 0, then W@ c 9’. (Hence W e c 9.) For f E ( E N ,

l l ~ l l ; I a;,71jlll 3 p = Wf,

zcith a constant independent if$

Proof: Let I f ( z ) l 5 yOl,(z), y = If/, , and let Irnj 5 k. Since Biml 5 pk (Fee (A.25)) arid Of,,(y) 5 Oi’,,(z), one obtains from (1.25)

B[”](z, q) 5 21m1/2n-n/41Bk,ei-,2,(~~ - q’)ef&,(z) .

I V e insert this estimate in the integral (1.24b). By (A.16), with #I = O:(x),

le. The Connection Between W and A

1.21. It remains to show that under suitable conditions W and A are inverse to each other. We first prove

22 V. BARGMANN

exists f o r all q, and (ii) the integral

existsforallw. Then (a) y = Wf isdejined, (b)g = Ay isdejined,andg = A(Wf) =f. The existence of y = Wf and of g = Ay follows from (i) and (ii), Proof:

respectively. We conclude from (ii) that

Using (1.19) and Theorem 1.3 one finds for the second form of the integral

dw) =/eW.Y(z) 44-4 = (e, , f ) = f (w) Y q.e.d.

On the basis of this lemma we have

THEOREM 1.6. (i) Iff belongs to Q: , p > n, or satisfies the hypothesis o f Theorem (ii) If y E P:, k > n + 1, then W(Ay) = y. (In 1.4a, then A(Wf) =f.

particular, (i) applies zf f E @', and (ii) applies if y E Yk.)

Proof: (i) Under the hypothesis, Lemma 1 is applicable. The proof of Theorems 1.4 and 1.4a shows that the integral S ( q ) exists and that [ ( q ) is a continuous function of polynomial growth. (ii) By Theorem 1.1 a, h = Ay E Q:+,,+ , and by Theorem 1.4, p = Wh(= W(Ay)) E As was just proved Acp = A(Wh) = h = Ay. Consequently it follows from Lemma 1 (Section 8) that rp( = W(Ay)) = y, q.e.d.

1.22. For the spaces Y and Cf we may summarize our results as follows.

THEOREM 1.7. A is a bijective mapping of Y onto (3, and W = A-l is its inverse. A and A-l are continuous in the topologies o f Y and E.

Proof: 1) By Theorem 1.2a, A Y c (5, but by Theorem 1.6, A Y = @. More precisely, every f E (f may be uniquely represented as f = Ay, y E 9, with y = WJ 2) Likewise, by Theorem 1.5, W e c 9, but, by Theorem 1.6, W e = 9'. Every y E Y may be uniquely represented as y = Wf, f E E, with f = Ay. 3) As to the continuity: by Theorem 1.2a, A maps the ball llyli: < c/uk into If/, < c, and, by Theorem 1.5, W = A-l maps the ball If I n + k + l < c/ah,, into

Ilvll; 6.


We note the inchion relation

2. The Spaces Ep and (5. Linear Functionals on @

The first part deals with convergence in C? and in (5. In the second part we derive the general form of a continuous linear functional on C?, which was mentioned in the introduction.

Some of the results in the first part hardly differ from the corresponding results of [6] (Volume, 11, Chapter 11, 92). They are, nevertheless, stated and proved-in a form appropriate to later applications.

2a. Convergence in @' and in

2.1. Preliminary remarks. (a) Some lemmas and theorems to be proved in this section depend on the properties of the Hilbert space 5 derived in Part I (Section 1 ). We mention specifically the inclusion relation

8,c iyc81=eo, A < l .

(b) We adopt the following terminology of Dunford-Schwartz [5] (p. 50). Let '9JI be a subset of a topological vector space 2. (i) The set of all finite linear combinations of elements g E 9.X is denoted by sp(9.X). (ii) '9JI is fundamental in 2 if sp('9JI) is dense in 2.

(c) The most important examples of fundamental sets in 5 are (i) Q (= sp( p)), the set of all polynomials, (ii) 8, the set of all vectors e, .

jd) A set 9N in 5 is called kinvariant if Q,9N c 9.X, A < 1. Both 5$ and 8 are A-invariant, the latter because are, = e,, .

(e) An elementfof 3 is called il-rep.ulur in (5, if

(ii) lim If - Q,fl, = 0 . A t 1

2.2. Convergence in @'. I t follows from the definition of the norm If 1, in Section 1.15 that

(2.1) If(4I 5 M ( r , f ) 5 PP(d I f l , if I4 5 r .

Convergence in Ep is defined as convergence in norm. A sequence (A> converges to f ( f , , f ~ ( 5 p ) if limi-w I& - f l , = O.* (A sequence converging to 0

We use no longer different symbols for strong and for point-wise convergence-as was done in Part I. The function spaces considered here give rise to various notions of convergence. In what follows it will be explicitly stated which limit is involved, unless this is clear from the context.

In general we write limi hi instead of lirn+- hi.

24 V. BARGMANN

is a “null sequence”.) I t is an immediate consequence of (2.1) that convergence in norm implies uniform pointwise convergence on boiinded sets.

LEMMA 1 . Let (hi} he a seqzierzce of functions in @‘ such that (i) their n o r m are bounded (IhiJp 5 c ) and (ii) the sequence ( h i ( z ) } converges for every z E C n .g Then the convergence is ungorm on bounded sets, the limit function h ( h ( z ) = limi-m h , ( z ) ) belongs to @‘, and /hip c.

Proof: By (2.1), Ihi(z)l 5 c P p ( r ) if 121 5 r, and it follows from the classical convergence theorems for analytic functions that on every set (21 < r the h i ( z ) converge uniformly to a holomorphic function h ( z ) . cOl , ( r ) if It1

Clearly lh(z)1 r, hence Ih), s c, and h E W.

THEOREM 2.1. Every is complete. Let { fi) be a Cauchy sequence in @. (For every positive E there exists an N ( E ) such that Ifi - f 9 1 p < E $ i , j 2 X ( E ) . ) Theit there exists a unique f E @‘ such that If - & I p -+ 0.

Since I ljJp - l ~ l p l Proof: Ifi - & I p , the norms l jJp converge and are bounded. By (2.1), I f , ( z ) -&(z)I < &€Jl,(r) if i , j 2 . N ( E ) arid IzI r, so that the sequence (fi(z)} converges for every z . By Lemma 1 the limit f belongs to EQ.

Set, for a fixedj 2 N ( E ) , hi =f, - A , i 2 N ( E ) . By hypothesis, Jh i (p < E for all i, and limi h, (z ) = h ( z ) = f ( z ) -A&). Using Lemma 1 once more we find that l f - f J p 2 E i f j 2 N ( E ) .

The following is a useful convergence criterion.

LEMMA 2. Let fi E E“, i = 1, 2, * * * , such that (i) ILl, y < co for all i, (ii) the sequence (A( z ) } convergesfor every z E C . Then, f o r every p < u, If -hi,, -+ 0 w h e r e f ( z ) = lim, fi(z). ( T h e f , need not converge in the norm 1.1, .)

Proof 1) Set gi = f -AI. By Lemma 1, Ifl, y, hence (pil, 2 y , and we use the following M(r, g,)

estimates. Let r > 0. Then 2y@,(r). 2) To prove lgJP -+ 0 if p <

r3;l(r’)M(r’,gi) 2yO;-,(r’) g 2yO;-,(r) = h(r) if r‘ >= r ,

O;l(r’)M(r’, gi) 5 +%f(r, gi) = c(i, r ) if r’ s r .

Hence IgJP s max (b(r ) , c ( i , r ) ) , and the scheme in A1 applies. In fact limr+mb(r) = 0, and, for every r, limi+m c ( i , r ) = 0 in view of the unifOrm convergence of the sequence (A>. Thus lgilp ---f 0, q.e.d.

COROLLARY. Proof: (See (e), Section 2.1). By Lemma 1 (Section 1.17), IQ,fi,

Let f E E‘. Then f is &regular in Ep ;f p < G.

I f l , = y if 2 < 1, and limLT1 (sZ,f)(z) = l imf( lz) = f ( z ) .

Here, and later on, it would suffice to require convergence on a suitably chosen subb(.t of‘ C:r , (see I, Section If). This refinement, however, will not be needed.


THEOREM 2.2. (rlppro.riinafion Theoreinj. Let YR tie a A-inrlariant jitndamentalset in 5. and Id f be ?.-regular in @'. For eveiy positive E there exists an element g E (sp (YJ; n E) j,,. ;&id I f - ,& < F .

Proof: For a suitable I. (< I ) , If - Onfl, < $ E . Choose ILf, 2.'' such that 1. < 1.' < A" < 1, and set ,u = A/,?' (< 1) . By Lemma 3 (Section 1.711, i2;,, f ' ~ 6,:. c 8. Since '%I is fundamental in 8, there exists h E sp(%l) such that, in \4ew of ( e ) , Section 1.16,

& IQ,,f- 4 0 5 IlQxf- hll < ;-

CPPO

(see Lemma 2, Section 1.17, for the definition of c P p o ) .

Then Chooseg = Q,,h (~sp('9Rj). (Note thatgE6,, c 0-,andhenceg~(sp(%) n@).)

IQJ- gl, = IQ,(Q, , f - h) l , 9 CPP" l Q n , j ' - hl, < '2.' 'Thus I f - sl,, 5 If- QJl, + I Q J - PIp < E, q.e.d.

2.3. Convergence in 0-. In the topology of @ a sequence {f.} converges to

Iff = A ~ J , fi = Ay, ( y , y L E 9) the condition is equivalent to limi lly - yiilt = f ( J J E (5) if, for every non-negative integer k, limi+a If - J J k = 0.

0, or Iim sup iOp(q) IiYmly(q) - iYmly,(q)l) = O i-m q

for e\rer)- I E N and every multi-index m.

Then lim,-., /f .-Alk = 0 for all k. 'The following is a simple criterion. Suppose If - & I i < E~ , and limi-.,f pi = 0.

(Proof: Fix k. If i 5 k, then I f - f , l , I f - f i l i < E ~ , 9.e.d.j

From the preceding results on (3, we now readily obtain the corresponding results on E.

THEOREM 2.3. 0 is romplete. Let {L.} be a Caiichy sequence in 6. (For euerv positice E and Every k E N there exists an -Vk(&) such that If, - J l k < E ij'-i, j >= ]ITk( P ) . )

Then there exists a unique element f in 0- such that lim, I j ' -,hlk = 0 f o r all k.

Proof The assertion immediately follows from the completeness of every Ek. I n fact, the hypothesis implies that, for every k, limi If = 0 with one and the same functionf'(defined byfi,t) = lim, fi(z)), which belongs to all @and hence to 0.

Lmim 1 .

Proof:

I f f E @, then f is A-regular in all 0,.

Fix p. Since j ' E @+l, the assertion follows from the corollary to Lemma 2, Section 2.2.

26 V. BARGMANN

THEOREM 2.4 (Approximation Theorem f o r (5). Let (3n be a 2-invariant fundamental set in 3. For every f E @ there exists a sequence {g t } c (sp((3n) n @) which converges to f.

Proof: For every i, f is A-regular in ei and, by Theorem 2.2, there exists a gz E (sp(93Jz) n @) for which I f - gzli < (i + 1)-l. Then lim, If - g,l, = 0 for every k , q.e.d.

The theorem applies in particular to the two sets 'p and 23 (see Section 2.1). It is also easily shown that E is a Monte1 space, i.e., that every closed bounded

set in @ is compact. (A set 23 in @ is bounded if it is bounded in every norm 1.1, ~

Then there are constants c, such that If/, 5 ck < co for every f E 8.)

THEOREM 2.3a. Every bounded inznite sequence {fi} in (3 contains a converging subsequence. (Hence every closed bounded set in (3 is compact.)

In view of the inequality (2.1) and the condition l f i l o 2 c,, one may select, by the Ascoli-ArzelB theorem, a subsequence { f i , } which converges point- wise (fi,(z) - f ( z ) ) for all z E C%. For every k , lfi,lK+l s c ~ + ~ , and it follows from Lemma 2 (Section 2.2) thatfE Ek and that lim, If-fi,lk = 0, q.e.d.

Proof:

2.4. Remarks on the functions e, . In what follows we shall need estimates

(a) A lower bound for lealp follows from the inequality for the norms of e, and of (ea+b - eu).

HILBERT SPACE OF ANALYTIC FUNCTIONS-. I1 27

and thus

(2.4a) rhk:lp I IblS

Inserting (2.3) we obtain therefore

(2.lh)

It is readily seen from (A.8) that this series converges (for every p ) . Thus the series (2.4) converges absolutely in every norm ( - I p .

(c) Finally, we introduce the remainder terms r:: and r t l by

(2.5)

For fixed p, fixed a and bounded Ibl (say Ibl s 1) we find from (2.4a), (2.4b)

ea+b = e, + r t i = e, + hbfd + r t i

2b. Functionals on (2 and Distributions

2.5. Preliminary remarks. By “distribution”-without any qualifying adjective-we always mean a tem..erzd distribution, i.e., a complex valued continuous linear functional, v, on 9.

The standard example of a distribution is the inner product

of a suitably chosen fixed function v with y . In analogy with (2.6) we denote the action of any distribution u on y by

(2.Sa) (0, Y } > Y E Y ,

i.e., { a , y} is a continuous complex valued function on 9; {u , y1 + yz} = { P , yl} + {u, y2>, and {v, hp> = i { a , y } for every complex constant 2. We write u = 0 if (11, y } = 0 for all y E 9’.

A distribution defined by an expression of the form (2.6) is called “regular”, and we use the same symbol-a-in both (2.6) and (2.6a).

CONTINUITY. In view of the linearity of u it suffices to require continuity a t y = 0. Thus there must exist a neighborhood of 0, for example, Ilyllfl < (1 + 1)-l

28 V. BARGMANN

such that I (v , y)I < 1 for all y in this neighborhood. This implies

(2.7) I{% Yll 2 c’ llvlllfl

for some positive c‘ and some k”E N. An inequality of this form, in turn, implies continuity of u.

If v ( q ) is a measurable function, and

(2.7a)

then (2.7) holds for the regular distribution u.

THE SPACE 9’. The distributions form a linear space Y’ with the following basic operations. (i) Addition: w = u, + v 2 is defined by (w, y} = { v l , y ) + { v 2 , y}. for every K E C, w = KV is defined by (ii) Scalar multiplication: {w, Y } = Y’).lo

L(c.l. fcl”. Remark. In the sequel we abbreviate “continuous linear functional” by

2.6. Functionals on (5. The mappingf = A y turns the c.1. fcl. (v , y} into a c.1. fcl. L( f) on (5 if one defines

(2.8) L(f) = {v , w} 9 f = A y E E .

Conversely, given a c.1. fcl. L ( f ) on (5, equation (2.8) determines a unique distribution u on 9.

Repeating the argument which led to (2.7) one concludes: for every c.1. fcl. L on (5 there exists a positive c and a non-negative integer k such that

(2.8a) [L(f)l S ~ ] f [ ~ f o r a l l f ~ Q .

Conversely, a relation of the form (2.8a) implies continuity.

{.fi} is a null sequence in Q, limi L( f , ) = 0. As is well known this criterion may be replaced by the following: whenever

If L and u are related by (2.8), then (by Theorems 1.2a, and 1.5)

I { v , y}I “ k lldlf (from (2.8a)),

(from (2.7)) . 2.7. Before determining the general form of L(f) we make two observations.

1) Let L, , L , be two c.l. fcls. on (5 such that L,(e,) = L2(e,) for all a E (En .

IL(f)I S % I , , I f l k ’ + ” + l

Then L, = L, . lo This definition of scalar multiplication follows [6] rather than [1 11 (where a is used instead of

ti). It is more appropriate to the formalism of this paper.


Proof: Set L, - L, = L. By hypothesis, L(e,) = 0. Thus the functional L vanishes 011 8 (see Section 2.1) and hence, by linearity, on sp(l2)). Since-by the approximation Theorem 2.4-everyf~ (5 is the limit o fa sequence (9%) c sp(%), and L(g,j = 0, we have L( f) = limi L(g,) = 0, i.e., L1( f) = L,(f) for all$

2) If h is an element of some @', then

(2.9) L(f) = ( h , f )

is a c.l.51. on E because IL(f)l sj!a--k,2n lhl, If I R if k > 2n - o (Lemma 1 , Section 1.18). By Theorem 1.3,

(2.9a) -

h(a ) = (e, , h ) = L(e,) . 2.8. General form of L(f). Let L(f) be a c.Z.fC1. on (5, and I L ( f ) I 5 c l f l k .

Guided by the last example we define -

(2.10) d a ) = L!%) > a E c : : . ,

and analyze the properties of the function g(a). 1) By equation (2.3),

(2.10a) Ig(a)l c lealR 5 c h p : ( a ) = clO;(a) . 2) We prove next that g(a) is a holornorphic function of a. It is sufficient to

show that, for any fixed a and sufficiently small ( b ( ,

(2.1 1) d" + b ) - g(a) = Qb) + W I Z ) , where I (b ) is linear in b. By (2.5),

For fixed a and IbJ 5 1,

IW::)I s c Ir:;lk = W V ) (by (2.5a)).

Furthermore, l ( 6 ) = L(hit1) is linear in b, because l ( 6 ) is antilinear in h::;, and hi': in turn is antilinear in b. Thus (2.1 1) is established.

-

I V e conclude from (2.10a) that g E E-k. 3 ) Consider now the c. l . fc l . L,(J) = (5, f), g being defined by (2.10).

Comparison with (2.9a) yields L,(e,) = g(a) = L(e,) for all a E ( E n . As was shown in Section 2.7 this implies that L = L, , Le., L(f) = (g,f).

-

W'e have, therefore,

THEOREM 2.5. Every continuous linear functional L( f) on E has the form

30 V. BARGMANN

- Here g is afixed element of some E-k and is uniquely dejined by g ( a ) = L(e,). converse(^, every expression o f the form Id( j ‘ ) = ( g , f )--g belonging to some E-”-dejnes a continuous linear functional on E.

2.9. We “translate” this result into the language of distributions. Let u E Y, and {D, y } = L( f), f = Ay.

(i) Since e, = Ax, (4, Section 1.13), g ( z ) = L(e,) = {a, xz>. (ii) The relation f (2 ) = ( A y ) ( t ) may be written as

(2.12)

Thus we obtain

THEOREM 2.6. Let v be a tempered distribution. Then g(z ) = {v , x,} is a holomorphic function in some E-k, and

(4 w> =I{., x .>{xz > w> d P n ( 4 > y € Y .

Conversely, every expression o f the form

dejines a tempered distribution, and {v, xz> = g o .

2.10. The &function. The distribution 6, (or d (q - b ) ) is defined by (8,, y } = y(b) for every b E R, . Denoting the corresponding function g by 8, we have

(2.13) J d Z ) = (6, > x,) = XZO = A ( z , b) ’

If b = 0, we write 6 and 8. Thus 8 ( z ) = n~n‘ae-22/2. Note that 6, E Go, and 16,10 = T - ~ / ~ ( in fact O;’(z) 16,(2)1 =

x‘ = x - b/1/2. The formula

where

(2.14) w ( 4 ) = (6, Y w> = (6, ,f> 7 f = A w ,

will be frequently used. It is a counterpart to equation (2.12) and expresses the equation y = Wf (Section 1.18) in terms of distributions.

2.11. Remarks on regular distributions. Let v ( q ) be a continuous function of polynomial growth, say, v E 9: . Thtn u defines a regular distribution

HILBERT SPACE OF ANALYTIC FUNCTIONS--II 31

(13) (2.611, the integral in (2.7a) being finite if k’ > n + p. The corresponding function q is given by g ; z ) = {Gz} = (AzJ) ( z ) , and by Theorem 1.1 a, g E D: c E-IPI . ‘Thus

(2.15, {u, y> = (A% Ay)

Conversely, if g belongs to 2: , ,u > n, or satisfies the hypothesis of Theorem 1.4a, then the distribution defined by {a, y } = (g, A y ) is regular, and the function v is given by u = Wg.

Proof: By Theorems 1.4 and 1.4a, Wg is a function of polynomial growth. According to (2.15) it defines, therefore, the regular distribution {Wg, y } = (A(Wg), A y ) . Since A(Wg) = g (Theorem 1.6), this distribution coincides with {% y ) , q.e.d.

2.12. The space (5’. The functional L-defined by L ( f ) = (g,f)-may be ident!fiefied with its generator g. Then the space (5‘ of all c.1. fcls. on %-the dual of Q-is simply the union of all Q?, k EN. In view of the inclusion property Q“ c QP, (r > p , it may also be described as the union of all EP, - co < p < CO,

and in other equivalent ways. In particular one may juxtapose

m m

@ = E k , (5’= u a h . k=- OD k=-W

EXTENSION OF THE MAPPINGS A AND A-I. The equation { a , y} = :g, A ~ I ) (for all y~ E 9) establishes a one-one correspondence between the elements of 9’ and those of (5’. We denote the mapping 9” + %’ so defined again by A, i.e., we set

also

I t follows from the definitions at the end of Section 2.5 that A is a linear mapping 9’ ---f Q’ and hence A-l is the inverse linear mapping (5‘ -+ 9”.

With this interpretation equation (2.15) holds by definition. Furthermore, the mapping A is an extension of the previously defined integral transform (equation (1 .7) ) ; for, it coincides with the integral transform for the regular distributions discussed in Section 2.11. Likewise A-l extends the definition by the integral transform W.

EXPLICIT FORM OF A. Let w denote either a test function (w = y E 9) or a distribution (wb = v E SP’), and set h = Aw. In both cases one obtains

(2.16a)

(2.16b)

32 V. BARGMANN

Here (2.16a) follows from the equations in Section 2.9, and (2.16b) from equation (1.23b), since h = Im (h,U,)~,, and (h, urn) = (w, pm>, by 5, Section 1.13.

2.13. Concluding remarks. We interrupt here the analysis of the c.1. fcls. on (3; it will be resumed in Section 4. In the next section we study the Hilbert spaces 3 p defined in the introduction.

Some of the new features are these. 1 ) Every element of 5" is 2-regular in 5". 2) For every p, C5 is a dense subset of 8". (This does not hold for any (30; see the examples in Section 3.16.) 3) The set of all c.1. fcls. on 5f is given by L ( f ) = ( g , f ) , with g E 3 - p . (The situation i s quite different for ( 3 p ~ From the Hahn-Banach theorem one may infer the existence of a non-vanishing c.1. fcl. L( f) on (!? which vanishes for all f E (3 and therefore cannot have the form L(f) = (g, f ).)

In connection with the last remarks it is of some interest to characterize the closure of E in Ep. The following lemma shows that this closure consists of allfwhich are &regular in e p .

LEMMA 1. For a fixed p, let 2 p be the set of all f which ure I-regular in Ep, and let RD (resbectiveb SUP) be the closure of E (respectively &", a > p) in &p. Then s i p = R'p = 2 p for every 0 > p .

Clearly Rp c Rap, and it suffices to prove 2 p c Sip, and R'P c 2 p . 1) Since !2,@ c E (corollary to Lemma 2, Section 1.17) every f which is A-regular in 83' belongs to the closure of i5 in Ep, i.e., 2 p c Rp. 2) Let f E Rap and let {gJ c eU be a sequence converging to f in Ep. Write

I < 1 .

Proof:

f - a f = (f - gi) + k i - fi,gz) + (a,& - fi,f 1 J

If - fi,f lp s 2 If - Sil, + I& - Rlgi I p *

Since IQnki - f ) l p s I & -fl, 2

Now limi If - gilp = 0 (by hypothesis) and, for every i, lim, 1 Igi - sZ,gzlp = 0 (corollary to Lemma 2, Section 2.2). Thus lim, t If - a, f I,, = 0 (see AI). Hence f E 2 p ,

or Rap c 2 p , q.e.d.

3. The Hilbert Spaces Bp 3.1. Basic definitions. Set, for real p and z E Cn ,

dp;(z) = O & ( Z ) dp,(z) = T-"OL~(Z) d " z ,

(3.1) G =jW) =SO& diU,(Z) (= j&?,,,, *

DEFINITION 1. Far every f E 3 the norm l l f l l , is giuen by


DEFINITION 2. gp consists of all functions f € 3 for which 11 f / I p < co. If g , f ~ zp, their inner product is

I V e note that

AN INEQUALITY.

c 8” if o > p, and that 5’ = 5.

Let g E S - P , f E 5”. Then (g , f) is defined and

(3.3) Ik,f ) I s IISllLP I l f l l p .

Proof: By Schwarz’ inequality,

3.2. Completeness of 3 P . We first derive a rough estimate for M ( r , f ) in

terms of 11 f ( I p .ll Set ,ur = min (rn/2&1(z)) . P Then 121 S T

Assume 121 5 r, and denote by v, the volume of a ball of radius 7 in C n . Applying the mean value theorem for analytic functions and Schwarz’ inequality, one obtains, for r > 0

f(z) = f ( z ’ ) d n z ‘ , l z ’ - z l ~ T

VT

(3.4)

In analogy to Lemma 1, Section 2.2, we prove next

LEMMA 1. Let {h,} be a sequence of functions in 50 such that (i) their norms are bounded (llht[Ip 5 c ) , and (ii) the sequence {h , ( z )] converges f o r every z E C n . Then the conwrgence is uniform on bounded sets, h ( z ) = lim, hd(z) is holomorphic, h E gp, and llhllp 5 c.

l1 This is a well known procedure. See, for example, [4], p. 5.

34 V. BARGMANN

Proof: By (3.4), M(r, hi) 5 v(r )c for all i. It follows that on every bounded set the sequence {hi(z)} converges uniformly to a holomorphic function h ( z ) . For every i and every positive r,

J; lhi(4l2 4 4 ( z ) 5 ll"lr; s c2 , 21 8 7

and hence, in view of the uniform convergence,

In the limit r + CO, we thus obtain llh/lp 5 c.

THEOREM 3.1.

Proof:

5" is complete, i.e., S P i s a Hilbert space.

The proof of Theorem 2.1 applies almost literally, on account of the preceding lemma and the inequality (3.4).

3.3. The action of the operators !2, . (Compare Section 1.17.)

LEMMA 1. rf 3, < 1, then, f o r any pair o,d, IIfiJIl, 5 c;,,, I l f l I , , with a positive constant c;,,, independent off.

Proof one finds

Using (1.21) and the definition of cAou, in Lemma 2, Section 1.17,

COROLLARY.

LEMMA 2. For all real p, (i) 5 p c Q, i f p > 1, (ii) QA c S P ;f 3, < 1,

For any pair u,d, !2,3"' c 5" i f 3, < 1.

(iii) .Rasp c 6,. $ A < A'. Proof This is a precise analogue of Lemma 3 in Section 1.17. It was

c

I t follows from Lemma 2 that Theorem 1.3 holds for the elements of any 8".

pointed out that its proof depends only on the relations Slap' c 5" and go c Q1 for 3, < 1. The latter is satisfied because %o = 5.

As an application we deduce from (1.23) and (3.3) the inequality

(3.5) If(4I = I ( % ,f>l s ll%Il-p Ilf /Ip > f f = 5 " '


3.4. The norms / /e , / / , . I n order to utilize this inequality we estimate the norms llenil,. From (2.2) we obtain, by squaring and integrating (with measure d' lz) ,

(3.6) ~llea\l,/~;(a))2 =Sre:c. + w)ie;(a)l2 d p n ( w ) , w = z - a .

Kow by (A.3),

(3.6a) K-'p 'e"! , ! (W) 6;(& + W ) / e ; ( U ) 5 K!P'6Up, (W) .

Hence, b y (3.1 j, - -

(3.6h)

.As for the norms lealp, we find, for fixed p,

K-!oIv'&p qa) 5 Ileal/!, 5 Kl+'q$l ~ " a ' p \ i .

(3.6ci

Since O:(a + w)/O: (a) -+ 1 as la1 -+ GO, and J dp,(m) = 1 , this follows from (3.6) by dominated convergence (see (3.6a)).

A closer analysis shows that, for fixed p,

(compare (2.3a)). Inserting (3.6b) in the inequality (3.5) we obtain

- (3 .7 ) I f ( 4 I 2 - W T , f ) 6 K ' P ' d 7 b " ' llfll,~:,(r) (1.1 5 9 f E

in close analogy with (2.1).

3.5. Connection with the spaces EP. The last inequality implies that I l f l l , , (see Definition 1 in Section 1.16) and hence that 3" c Ep.

O n the other hand, @p+'+P c 5 p if > 0. I n fact i f f€ E', 0 = p + n + b, lfl, 5 K;,'

then I f (z)I 5 Ifl,O'b(t). Thus, by (3.2) and (A.10),

IIJIIE 2 ~ f ~ : r - " ~ ! t n - t f i ( z ) d n z =j:zn-z f i , zn I ~ I ; *

Thereby we have proved

36 V. BARGMANN

It follows that on E the system of norms I l . l l P is equivalent to the original system 1.1,. In addition,

m +m m + m

e : = n ~ k = n p , g ' = u g - k = u t . Y k * k=O k=- CG k=O k=- w

3.6. The orthonormal basis ( u k ) of 8". For any f E 3 we set, in accordance with equation (1.6b) in Section 1.7,f(z) = limz,mfi(z), where

Note that, by the multinomial theorem,

(3.8a) 2 u,ou,(z) = ( a * z)S/s!, 2 1u,(2)12 = IZ12S/S! . lml=s Imj=s

Let

(3.9) n

then ( u r n , urn,)p = 7&,.(co). v = 1zI2, and

In order to compute these inner products set

where da denotes Euclidean measure on the unit sphere. We find

Introduce, for real p and positive integral I ,

(3.10)

= J$IJj = J:/I'(A) , <;= 1 .

Since u, are orthonormal on 3O, T",,.(co) = +ymm. r ( n + s) = a,,,, i.e., Zymn, 1 = d,,,,, /r(n + s), s = (m(.

(3.1 lb) 7&,,(=)) = ~ m m , qpml > qps = {p,+, , q; = 1 .12

7; = (1,

Hence

(3.11a) T:,. = 0 if m # m',

l2 Note that the definitions of 17: in equations (3.1) and (3.1 Lb) coincide. Both amount to


It follows that the functions

(3.1 lc) u; = (qf,, )-%-

are orthonormal on @.

COMPLETENESS OF THE SYSTEM (uk). Let f = X u,u, E 8”. Then

J um(Zlf(z) ~ P P , ( Z ) = 2 TLm’(r )am, = T ~ m ( r ) ~ m 9

21 5 r m’

by (3.9) and (3.11a). Hence, for T + 00, (u , , f ) p = q P m cc, . This implies the completeness of the system u$ ; for, the only element in 5” orthogonal to all u : ~ i s f = 0 .

As a result, iff = C umum and f’ = C uhum E 3,

(3.12)

- (3.12a) ( f , f ’ ) p = 2 rfml 9 Aft E 8 p .

m

(Equation (3.12) may be interpreted as follows. Either both sides are infinite, or both have the same finite value, in which case f E 5”)

According to equation (1.23), one may write instead of (3.12a)

(3.12b) ( f j f ’ )p = 2 qp,&f) u,>(fl, ,f‘) > f, f’ E sp m

Denote by ’p, the linear set of all homogeneous polynomials of order s. Its dimension-the number of multi-indices m for which Im( = s-equals

- 1 + s (3.13) Y, = dim p, = (“ n--l ) , vg- (n + s)n-l/(n - l ) ! for large s .

00

In any sp, the ‘ps are pairwise orthogonal, and iJP = 2 0 13, . Ifp, p’ E Q, , then .%=I)

Furthermore, for any f E B P ,

(3.13b) Ilf -fdp + o as I - + 1

(see (3.8)), and fl E 2 0 V S . s=o

13 If the mapping A is applied to the functions u in Grossmann’s space S@), then Au =f is (x,Jz. (See [a], equation (3.3).) holomorphic, and i;u!/’ = Zm

38 V. BARGMANN

3.7. The coefficients qf . Before continuing the analysis of the spaces 5” we derive some inequalities and asymptotic expressions for the coefficients 7; . The results needed in the following discussion are summarized at the end of this subsection.

We consider first J5 and 2;pn. Setting w = 1 + u we obtain from (3.10)

m

J t = tl wP(w - l)A-l ecW dw I e wPtA-l e-W dw . (3.14) -r From the definition (3.10) we find

(3.15a) JP, + Jpn+l = J$+l , I;$ + nl;.,,, = SP,+’ .

.{+I (4 + dxp,,, (u)/du = nap, (u ) + p.P,7;:‘,(0,

[%+I = l;: + pt:?l*

Integrating the identity

from 0 to 00 we obtain

(3.15b) J5*1 = 1 4 + PJp;; 9

INEQUALITIES. For non-negative p, by (3.10) and (3.14) ,

uP+A--l e-* du J5 < e mP+A--l e-W dw i.e., 1 la (3.16a) T(p + A) 6 J p n < e r ( p + 1) , p 2 0 .

Similarly, for negative p, if p + A > 0,

(3.16b) Jpn < F(P + 4 7 p < o , p + 1 > o . ASYMPTOTIC EXPRESSIONS FOR FIXED p AND LARGE 1. We assume p # 0 (the case

p = 0 being trivial) and start from the inequalities (for positive u) :

(9

(iii)

( 1 + u ) P 5 up + p ( l + u y - 1 ,

(1 + u ) P < up + pup-1,

(1 + u)P > u p + pup-1,

(ii) p 2 1 ,

O < p < l ,

p < o . From (3.10) and (3.16a) we obtain, respectively,

(9 Jt 5 r(n + p) + pJ5-l < r(1 + p) + e p r ( p + A - 1) , (4 J$ < r(1 + p) + p r ( 1 + p - 1) , (iii) J{ > r(a + p) + + - 1) .

(In the third case, A + p > 1 is assumed.) Hence, from (3.16),

p < 0 , p + 1 > 1 .


Thus for fixed p (including p = 0), by Stirling’s formula,

(3.17) J; - r(i, + - my,q , 2;; - P , for large A ,

ASYMPTOTIC EXPRESSIONS FOR J f p ( A fixed, p + + co). If A = 1,

= e r ( p + 1) + O(p-l) . 1 J { = ( /omwPfi-l e-W dw - wP+L-l e-W dw l If I 2 2, then (w - l)i-l 5 wi-l - ( A - 1 ) ~ ~ - ~ , w > 1. Thus, by (3.14) and (3.16a),

Hence for all 1 (including A = 1)

(3.18) J{ - e r ( p f 4 - e r ( p ) p i , t‘% - eI’(p)pi /Y4 , p + + m .

To find the corresponding expressions for J;” we use the inequality 1 > e+ > 1 - v, u > 0. Thus by (3.10), ifp > A + I ,

[(l + v)-Pv’I--l dv > J p > (1 + u)-P(vi--l - u a ) dv, 1 B ( I , p - A) > JiP > B(A, p - A) - B(A + 1, - 1 - 1)

(see (A. 14)). Hence

(3.18a) J;;P - B ( A , ~ - n) -f-Lr(t), 2;;p - p - l f p - , + m .

Note that ( f { ;P - e I ’ ( p ) / r ( A ) , p + + co. SUMMARY. For the coefficients 7: = [E+8 the main results are:

> (a) q: < 7: if p > , (b) : 7; as p 0 ,

(d) c i 5 qt(n + s)+ 5 c,”,

(f) q; z up + n)mn) if > o , (h) 7: s (1 + IPI)TJ:+l.

(c) for fixed p , 7: - (n + s)P as s-+ 03,

(3.19) (e) 1 5 q,pri: 5 cp >

(9) V t + (n + J)VL = ~t+1>

Comments. (a) follows at once from (3.10), (b) follows from (3.15b), and (c) from (3.17); (d) follows from (c) with some positive constants c i , cb . The first part of (e) follows from (3.3), withf = g = u, , Irnl = s, the second part from (d),

40 V. BARGMANN

with cp = cz,c6 ; (f) follows from (3.16a), (g) from (3.15a). (h) is trivial if p 2 0 (by (b)), and it follows from (3.15b) if p < 0, because then q: = qR1 +

It is easily shown that, for positive p, suitable values for c; , ii in (d) are: C; = P ( p + n) /r (n)nP, c: = e if p 2 1, and C; = 1, c; = e r ( p + n ) / r ( n ) n p if

IPI v z : < (1 + lPl)v:+l (by (a)).

O < p < l .

3.8. T h e principal vectors e,P and t h e functions F ~ . I t was shown in Part I (Section lc) that an inequality of the form (3.7) implies the existence of "principal vectors" eP, in %p such that, for every f E gp and every a E C,

(3.20) (e; dp = f ( 4 . Using equation (1.9b) in I, and equations (3.8a) and (3.11~) of the present section, we find

m s=o /m/=s S

m

(3.20a) e,(z.) = E p @ ' z) Y E p ( 0 = L: I;" /(s! vz) Y < € @ . s=o

Here sp is an entire function, so([) = ec, and el = e, .

f = 0 is orthogonal to all eP, . ESTIMATES FOR ~ ~ ( 5 ) .

Clearly Bp, the set of all principal vectors e; , is fundamental in 3 P because only

Inserting f = et in (3.20) we obtain ( e z , e i ) p = et(a) = ~ ~ ( 6 . a), in particular ]le;llE = ~ ~ ( 1 . 1 ~ ) . Combining (3.20) with (3.5) we have, therefore,

%(a) = (e; , ela12 =

1le;ll; =

s IIe",lp lle,llp >

= ( e a > e:) s lleall-p lle;llp

Hence eiala/lleallp S lle:llp S Ileull-pY and by (3.6b), (3.6d)

(3.21)

(3.21a)

(l/Yp)e",,(.) 9 &p(la12) 5 YpO?,, Y yp = K W j l t I ,

sp(la12) = (1 + o(e! , (a) )V, , (a) for large la1 . Since I&,(()\ .s,(I{l), we have

(3.21b)

The remainder of Section I c, d in Part I may be applied to 5" without change. There exists, in particular, a reproducing kernel X p ( w , z ) = cp(w . 2) on gp.


3.9. Further properties of gp. LEMMA 1. IfA < 1, then ~ ~ ~ J ~ ~ p 5 ! [ f l l P , and, f o r ezlery f E gp,

lim 11 f - Q t , f l l p = 0 . a t1

Let f = ?; M , U , . Then //sZ, f 11; = xln qplnl Azm lccm[2, and Proof:

l l f - QAfIl; = z: ll$l (1 - Llrnl)2 l % I 2 , m

from which the assertion follows.

2.1). As a consequence we have the following stronger approximation theorem. Thus every f E BP is A-regular in gp (in the terminology introduced in Section

THEOREM 3.3. Let !Dl be a R-invariant fundamental set in 8, and f E r . There exists n sequence {g,} c (sp(fm) n E) such that 1imi-= /!gi - f [ I p = 0.

Proof Since f is necessarily 2.-regular, the proof of Theorem 2.2 applies without any change.

LEMMA 2.

Proof:

In the topology of 8p, (3 and all s", a > p, are dense in gp. By (3.13b), the set $ of polynomials is dense in g p , and the assertion

follows from the inclusion 9 c ($ c 5" c 8 p .

LEMMA 3. Let f = C Q,U, €3. (i) I f f E 3p, then lccJ2 5 y ( n + I rnl ) -p f o r some positive y . (ii) I f a > p + n and 1xrnI2 5 y ( n + [ml)-', then f E 3 P .

Proof (i) From (3.12) and (3.19d),

lam12 2 l I f l l ~ / q ~ m l 5 C ' p ' I l f IIiCn + Irnl)-P.

m

(ii) Here l l f ] ] : 5 y zm v,qf(n + s)+, where v s = dim ps (see (3.13)). Asymptotically, v,$(n + s)-" - const. (n + s ) P + ~ - ~ - - ~ , so that iljIli <

( n + Irnl)-' = y s=o

i fa > n + p.

LEMMA 4 (Weak Conuergence). The following conditions are necessary and sujicient for the weak convergence o f a sequence {A) c 8": (i) For every z E C n , {A(.)} is a convergent sequence. (ii) For all i, / / J / l p y for some positioe constant y .

1) Necessity. For a weakly convergent sequence {jJ the norms are bounded, and (e: ,A),, = f i ( z ) converges for every z . The (weak) limit f is determined by f(z) = (eg ,f)" = Iimz (ef , f i )p = lim, f,(z). 2) Su.ciency. By hypothesis, (9, converges for all g (= ef) in %P, hence-by linearity-for all elements of the dense set sp( W). Since the norms are bounded, (g , f i )p converges for all g E 80.

Proof:

42 V. BARGMANN

COROLLARY. If, in addition to (i), (ii), limi llfi / I P = Ilfll, , where f ( 2 ) = lim, A(z), then lim, Jlf, -fl], = 0, i.e., {fi} is a strongly convergent sequence.

3.10, Connection between different %P.

LEMMA 1. zf 0 > p, the norms II-Ila and II.I(,, are inequivalent on (3, i.e., for a suitable sequence {fs> C @, lim, l l f s ~ l u / ~ l f s l ~ P = a.

Proof: SetJv = (22)‘. By (3.13a),

llfsllu/llfsllp = [q;s/q;s11’2 - ( n + 2 s ) ( u - P ) / 2 - ( 2 S ) ( 4 1 2

(compare (c), Section 1.16).

LEMMA 2. Let f = X amurn E 5“. I f p < o - n, then the series I: a,u, c0nverge.r absolutely in the norm o f Sp , i.e., Z larn/ JIurnlJp < a.

Proof: Let y = zm Ia,l lIu,(lP = zrn (qfm1)112 la,!. Then

As in the proof of Lemma 3, Section 3.9, we write the last series in the form

2 vsq:/q: . Asymptotically, vsq:/qz - const. ( n + ~ ) ~ + ~ - ~ - - l , and the series

converges if o > p + n.

m

S=O

COROLLARY. rf f = Z amurn EC?, then the power series 2 umum converges absolutely in the norm of every 8”.

LEMMA 3. Let f E E. I f f # 0, then, for positive p, 1l . f I j : 5 qiy with a suitable constant y > 0, and jl f j jP -+ 00 as p + co.

Proof:

It is easily shown, from the estimates in Section 3.7, that

By (3.19b), qf 5 qg for positive p. Thus IlflI; 2 qg 2 JamI2 =

Ilfll,, = 0 71: llfll: (forf= 1, llfll; = 71;). BY (3.19f), q; - a as p -+ a.

iff belongs to any 5“. (The corresponding assertion for the norms I . I P is false.)

3.11. The inclusion map M; . The connection between 8“ and 5p, o > p, may be studied with the help of the inclusion map Mz defined by

M;f =f , J’E v, M; f E 3”. Since llM;fljp = I l f l l , 5 J j f / l o , M; is a bounded linear mapping of 5“ into Ef’. In terms of the orthonormal bases u$ and UL of 3“ and 8” (see (3.1 lc)),

Mpu; = a I ~ l u , U P P , = [ q 3 7 , ” ] 1 ’ 2 .

HILBERT SPACE OF ANALYTIC FUNCTIONS--11 43

Consequently, M; has the polar decomposition = CAT; uniquely defined by

N:u& = (x~‘;$u;, VUuu p m = UL . Here .V; = ((L14;)*M~)1’z is a selfadjoint positive definite mapping of 5” into itself, and Vz a unitary mapping of 5“ onto 5”.

\.Ve draw the following conclusions : (a) The incluszon map M; is compact (= completely continuous).

Proof It suffices to show that N; is compact. Now N; has a discrete rpectrum,

(b) Thep-thpower of N; , p > 0, has a jinzte trace ;f and only zfa > p + 2nlp.

Proof: For large s, V , ( Z ~ ~ ” ) ~ - and the eigenvalues converge to zero since utpp N ~ ( p - ~ ) ’ ~ for large s

m The trace of ( N ; ) p equals 2 V , ( E ~ * ~ ) ) ” .

s=o

const and the series converges if and only if 2n + p ( p - G) < 0. It follows from (b) that (i) iW; and N; are in the Hilbert-Schmidt class ( p = 2)

if a > p + n, (ii) ill; and N; are nuclear operators ( p = 1) if a > p + 2n ([7], Chapter I, $2.3).

As a consequence we have ([ 71, Chapter I)

THEOREM 3.4. (% = n ijk zs a nuclear space.

In addition it follows from (a) that every weakly convergent sequence in 5” is strongly convergent in 5 p if p < 6. In combination with Lemma 4, Section 3.9, this implies: Let {f,} c ’$’ be a sequence such that (i) the norm / / f c l I u are bounded, (ii) the sequence { f , ( z ) > converges for every z E C . Then lim, / I f -Allp = 0 f o r every p < a ( f ( z ) = lim, f , ( z ) ) . (Compare the analogous Lemma 2, Section 2.2.)

m

k=O

3.12. Linear functionals on gp. 1) Let (gp)’ be the set of c.1. fcls. (= continuous linear functionals) on 80, and define the norm of a functional L (E ( S P ) ’ ) by

IlLll; = SUP IL(f ) l . llfl’pS1

According to Riesz’ theorem, L( f) has necessarily the form L(f) = (h , f ) , with a uniquely determined element h = 2 ymum E S P , and llL]]; = IlhIl,. Thus

L( f) = Cm qfml if f = Z cxmum E 8 0 . Since L(u,) = qrml ym and urn =

(urn , f ), we have

(a) I I L I I ~ = iIhIIp = (2 (qfmi)-‘ IL(um)12)1’2, In

(b) L(f) = 2 L ( u m ) ( u m ,f) 3 2 IL(Um)(Urn ,f)I S IILII; I I f I I p . m m

44 V. B A R G M A ”

2) A second form for L is suggested by the inequality [(g, f ) l 5 l/gll-p llfll,, (see (3.3)), which implies that L(f) = (9, f) is a c.l.fil. on B p for any fixed g E 8 - p . Here L(urn) = (g, urn) (and alsog(a) = L(e,)).

--

We assert that every c.1. f c l . on f j p may be expressed in this form.

Proof -

Let L E (Sp)’. Define g = zm L(u,)u,, and L,(f) = (g, f ) . Then llgll?p = zrn qrki lL(u,)l2, and since ($)-l 5 q ; P cp(q:)-l (by (3.19e)) it follows by comparison with (a) that

This shows that g E 8-p . (9, urn) = L(urn), i.e., L, = L, and L(f) = (9, f), q.e.d.

Hence L, E ( j j p ) ’ , and by construction, Ll(urn) =

Inserting L(u,) = (g, urn) in (b) we thus obtain

THEOREM 3.5. Every continuous linear functional on 3” has the form L( f) = (9, f } with unique g E 5 - P ( g ( a ) = L(e,)) , and, for every g E S-P, L ( f ) = (g,f) is a continuous linear functional on The bound l\Li/; satisjies the inequalities IlLIlI, 5 llgil-p d c , , llLllb . Iff E S P , g E S - P , then (9, f) is gioen by the absolutely converging

-

-

~en’es zrn k, U r n > ( u r n ,f > *

Additional remarks. 1) Note that (80)’ is a Hilbert space based on the norm (a). 2) If 0 > p, then 5“ c @’; hence (SP)’ c (5”)’. Likewise, IILII; 2 llLllk . 3) We have established a one-one correspondence between the elements of ( S P ) ’

and those of 8 - P : L( f) = (9, f), and g(a) = L(e,). If scalar multiplication in ( j j p ) ‘ is defined in the same way as it was defined in 9‘ (see end of Section 2.5), the correspondence L t , g is linear. We may then identify L with g and say that ( S P ) ’ and 5-P contain the same elements and have different, but equivalent, norms (see (3.22)).

-

For a later application we note that

Proof: Let p = min (p , , p2). Then g, E SP,j = 1, 2, and QAgj EE c 8 - p (corollary to Lemma 1 , Section 3.3). Thus both sides of‘the equation are defined, and by Theorem 3.5 both sides are equal to zrn A’rn1 (gL , urn)(urn , g2) , q.e.d.

3.13. Integrals on SP. An integral of the form

HILBERT SPACE OF ANALYTIC FUNCTIONS-I1 45 -

gives rise to the linear functional I (h) on 8”:

More precisely, letf, be a family of vectors in 8”. Here T varies over the subset B of a topological space on which an integral with the usual properties is defined. (In the simplest case T E [P,, and B is a finite interval. In a later application (Section 8.6), B is a compact group, and d7 denotes invariant (Haar) integration on B.) If the integral I (h) in (b) exists for every h E 8, and II(h)l y IlhlI, for some constant y, then I ( h ) is a c.1. fcl. on 8” and uniquely determines an element g in %”-designated by the integral in (a)-of norm llgll, s y.

simple criterion for the existence of the integral (a) is the following: (i) For every z , f 7 ( i ) is continuous in T ; (ii) l[Allp a(T), where a(.) is continuous and

j;(Tj dT < co. (If B has finite measure, it suffices to have l / f , l / , a,, , a

constant.)

Proof: By Lemma 4 (Section 3.9) on weak convergence, the conditions imply

-

that ( h , f , ) , is continuous in T for every h, and by Schwarz’ inequality n

For h = eP, one obtains g(z) = jBf,(z) dr.

It follows easily from Section 3.12 that under the same assumptions, for every h’ E s - p ,

Finally one may define an operator integral

c c

iff, = X7 f satisfies the required conditions for every f E 5,. 3.14. Functionals on (5. On the basis of the results of Section 3.12 it is

quite easy to rederive the general form of a c.l.fc1. on (5 (Theorem 2.5). If L is a c.l.fc1. on (5 one concludes as before that, for some k E N and a positive constant c, I L ( f ) i 2 c [ i f l l , , f~ 0.. Since (5 is dense in 5k (Lemma2, Section 3.9), L may be extended-by continuity-to a c.l.fcl. on gk. Then it follows from Theorem 3.5 that, for everyfg (5 ( c sk), L(f) = (9, f )-with a unique g E %-k.

46 V. BARGMANN

In combination with Theorem 3.5 we obtain the following translation into the language of Y’ (since u,,, = Apm by 5, Section 1.13) :

THEOREM 3.6. Let u E 9”. Then (v, y } = En& (u , pn,>{ym, y ] f o r y E 9. The series converges absoluleiv, and zm qrAl l {u , 9;.,n}12 < cc for some k E N . Conversely, if znl qi,$ (8,,12 < a f o r a sequence ofcodicients P m , then (v , y } = zm Pm{P)rn, y> is a tempered distribution, and { u , pm} = Pm .

As was already mentioned in the introduction, the criteria for tempered distributions stated in the theorem coincide with those indicated by L. Schwartz [ l l ] (Volume 11, p. 118) since cL,(n + 2 772, 5 c z k ( n + Iml)-” A similar remark applies to the criteria for test functions. In fact, y E Y if and only if Ay = f E gk for all k , i.e., if and only if

2 (n + Iml), l{pm , y>Iz < 00 for all k E N , m

since { p m Y Y> = (urn 3 f>. 3.15. The Gel’fand triple E c go c @’ ([7], Chapter I, 54.2). I t is clear

that the spaces E, 5 O , E‘ form a Gel’fand triple (or “rigged” Hilbert space) @ c H c @‘ if the following identifications are made: 8”- CD,, ?Jo ++ H (here k E N; k > 0). Then, by definition, (gk) ’++ @-, , k > 0. Since (5L)’ may be identified with %-h (see the remarks following Theorem 3.5) one obtains the correspondence 8‘- ale for all non-vanishing integers.

3.16. Additional remarks on the spaces (5’’ and 8,. We first consider a family of functions which separate different (5’’ and Z P . Let z = (z, , - * , zr l ) , and set

--oo < 7 < a, f r ( z ) = & , / 2 ( 8 3 ) 9

where E,(<) is the function defined in equation (3.20a) : (i) f, E 5” zfund on@ zyp < T - g. By (3.20a),

00

J r ( z ) = Z: ur,szY/ do! 9 IKII; = 2 qgs IK , ,~ /~ > s=o 9

where c ~ ~ + ~ = ( P s ! ~;’~)-ldm. The assertion follows from Stirling’s formula and the asymptotic relation qf - 5P.

(ii) f, E (5, i f a n d onb i f p We turn next to the relation of (5P and s i p (the closure of (5 in W; see Lemma 1, Sec-

(iii) f p is not I-regular in 0”. hlore precisely, if g E 6, , I < 1, then If, - gIp 2 2P’2 .

Proof: Consider real points x = (xl , 0, * * , 0 ) . Now

T (use (3.21b)).

tion 2.13). We assert

If, - gl, 5 lim o;l(xI) If,(., - S ( 4 l ’ x , - m

Since gE GA , lim O;’(xl)g(x) = 0, and, by (3.21b), lim S; l (x l ) fp(x) = 2 P ’ 2 , q.e.d.

R P (see Lemma 1, Section 2.13). Thusf, has, in W, a distance 5 2 P ’ 2 from all A-regular elements and hence also from

HlLBERT SPACE OF ANALYTIC FUNCTIONS-I1 47

In fact, R P is a very small part of 0,. Since the polynomials are dense in 0, R P is separable; V, on the other hand, is non-separable. For a proof it suffices to exhibit an uncountable family of elements h in 0.0 such that Ih - h’l, 2 1 if h # h’.

I f p = 0 such a family may be constructed as follows. Set for every complex number w of modulus 1, h,(z) = eozf’z. Then h, E E0, and it is easily seen w # UI’.

that Ih, - h,,l, 2 1 if

(For p # 0 one may choose h,(z) = 2-P‘2k,,/,(wt:/2), where m

m= 0 /i,(n = 2 m r ( ~ l + m + I) .)

4. Convergence in (5.’. Linear Operations on (5. and E’

In this section the analysis of the spaces E and (5’ is continued. The treatment of (5’ is based on the simple notion of weak convergence, which is adequate for our purposes.

Some general results about continuous linear operations on C? and on (5’ are derived. The class of “properly bounded” operators is singled out for a somewhat more detailed study because it contains most operators that are of interest in distribution theory.

The last subsection deals with the operators V, , Vu :

(V, f) ( 2 ) = ec.(z-c /2 l f (z - c) , ( V U f ) ( t ) = f ( U - l z ) ,

where c E Cn , and U is a unitary transformation on C . These operators were already introduced in Section 3 of Part I, where their action on the Hilbert space 8 was investigated. The analysis given here, though very similar, exhibits some new features. As will be shown in Section 5 many standard operations on distributions-such as translation, differentiation, Fourier transform-correspond to operators of this type or may be easily derived from them.

4.1. Convergence in (3‘

DEFINITION (Weak Convergence). A sequence {g,} c (5.‘ is convergent if the (numerical) sequence (g, , f) convergesfor every f E (5.. (Equivalently, f o r v, = A-lg, , y = A-lf, a sequence {v,) c 9’ is convergent ;f the sequence (vi , y ] coniwges f o r every y E 9.)

A null sequence is a sequence with limit 0. Convergence criteria may be derived from classical category arguments. The

following simple version will suffice. (It is a special case of Theorem 1’ in [7], Chapter I, $1.)

PROPOSITION. Let {p@( f ) } be a fami ly of continuous real-i’alued funcfionals on (5.

which are non-negative and absolutely convex (i.e., p,(Af) = 111 p a ( f), p a ( fi +f2) 5 p,(f,) + pa( fi)). I f p ( J ’ ) = supa p,(f) is Jinite for every f, there exists a closed b a l P

l4 For the radius Y of an open or a closed ball (formed with respect to any norm) it will always be assumed that 0 < r < ao. (Thus in the above proposition 0 < ,!? < m.)

48 V. BARGMANN

(If I k p, k E N) on which p ( f ) 1. Htnce, for all CI and all f E (5,

The norms I . I k may of course be replaced by the equivalent norms l l . l l k .I5

THEOREM 4.1. A sequence {g,} c 0.’ is convergent ;f and only ;f (i) all g, belong to some j x e d 0 . P (respectively 3 P ) and (ii) liml lgi - gl, = 0 (respectively limi jigi - gl/, = O ) , where g(z) = lim, g,(z).

Proof Necessity the condition. The proposition applies to the functions P i ( f ) =

I(gi, f ) l . (Since p i ( f ) is convergent, p ( j ) = sup, pi( f) is finite for every f.) Thus 1 (gi , f)]

It suffices to prove the assertion for the norms I . I p .

c Iflk for some k E N, and therefore

(see (2.10a)) or Igi[+ 5 c1 for all i. In addition, gi(z) = (gi , e,) converges for every z. I t follows, from Lemma 2, Section 2.2, that limi Ig, - glP = 0 for every p < --k.

Su&ciency o f the condition. The assertion follows from the inequality

which holds if I > 2n - p (see (1.22a)). The inequality also implies that lim, (gi , fi) = (g, f) if {g,} and {f,} are sequences converging, respectively, in @’ and 0. to the limits g and$

Remarks. 1) The proof shows that the condition may be replaced by the following: (a) For some CT the norms lgila are bounded, (b) the sequence {g , (z ) } converges for every z . 2 ) The theorem also establishes the completeness of (5’ (equivalently, of Y’) with respect to weak convergence.

EXAMPLES. For any g E El, 1) 1irnlAm g , = g (see (3.8) and (3.13b)), 2) lim, + Q,g = g (Lemma 1, Section 3.9). Note that the elements of both sequences belong to (5.

4.2. Analytic functionals. For a later application (in Section 8) we include a brief discussion of functionals which depend analytically on some parameter 7.

l5 In the remainder of the paper the definitions and the main results wiil usually be formulated in terms of either the norms I . l p or the norms /I. Ilp-the choice being a matter of convenience-since the equivalence of the two sets of norms has been established in Theorem 3.2.


DEFINITION. Let D be a connected open set in some Ern . The mapping g,( q E D -+ g E E’) is anaLytic (is€., holomorphic) on D ;f, f o r every f E (5, (g , , f) is an analytic function of 77 (on D).

TIIEOREM 4.2. (i) For g, to be analytic on D it is necessary and suficient that (a) for eziev1 rotnpact subset M of D there is an integer k and a positive constant y such that lg,,l-k y ;f 7) E M, (b) the function g(q, z ) = g,(z) is holomrphic on the Cartesian product D x S, (us, g is holomorphic in the m + n variables q ) 2 ) .

;ii) For every j , 1 2 j 5 m , (@g/aq3 , f) is a continuous linear functional analytic on D, and ‘.L a,./+,) = (ajar,) !f, 5).

Proof: Necessity o f the conditions. (a) For a compact Mletp,( f) = [(g, , J ) I , and p ( f ) = sup,&(f), q E hf. Since (f, g,) is analytic in q, p ( f ) is finite for everyfs @. -4s in the proof of Theorem 4.1 we conclude from the proposition that l!g, , f , l 5 c I f l k (for suitable k and c) and that therefore Igrll--k y if q E 12.1. (bj For fixed r), g ( r , z) = g,(z) is analytic in z , and for fixed z , g(q, z) = (e, , g,,) is analytic in q (if q E D). If q E 111 and 121 5 r, then Ig(q, z)I is bounded b y “/:(Y), and it follows from an elementary version of Hartogs’ theorem that g(.rj, z) is holomorphic on D x (C,.

Su$cienv ofthe conditions. Let qo E D. There exists a closed ball M : (17 - qol 5 b) contained in D and hence two associated constants k, y . Set 1 = 2n + k + 1 . For any f E B,

is analytic in q (if I?) - qol < /3) because the integrand is analytic in q and uniformly bounded by the summable function y If I l O12n-_1(z) (see I, p. 191, Theorem C) .

Since j ~ ( q , .)I B, it follows that lag(qo, z)/aqJ 5 /?-1#i(z,), ix.. ag/aq, E e’ for every point 710 E D. Furthermore, the derivatives of the last integral are

y€J;(z) if 1q - qol

if Iq - ylol < which proves the last part of the theorem.

(I, p. 191, Theorem C). For every f E (5 these are analytic in q,

4.3. Linear functionals on (5’. I n accordance with the program to treat

.’I linear functional L ( g ) on @’ is continuous if it maps every null sequence (g,) c (5’ the space E’ on the basis of weak convergence we define:

into a (numerical) null sequence {L(gJ} .

50 V. BARGMANN

An example of a c.l.fcl. L(g) is provided by

where f is a fixed element of (5. (Let {gi ] be a null sequence, so that lim, llgill-k = 0 for Some k. Then ILkJI = ICf, gJ 5 k i 1 i - k I l f ilk-see (3.3)- and limi L(g,) = 0.)

We show next that every c.1. $1. on CZ' has this form.

THEOREM 4.3. Every continuous linear functional L(g) on (5' has the fo rm L(g) = (f, g ) with a uniquely determined f E (5.

- Proof Let f ( a ) = L(e,), and consider the restriction of L to any %-k. If

limi llgi[l-k = 0 for a sequence {gi> c B-k, then lim, L(g,) = 0-because {g,} is a null sequence in (5'. Hence L defines a c.1. fcl. on the Hilbert space 8-k, and it follows from Theorem 3.5, with f and g interchanged, that f E 3k and L(p) = (f, g) for all g E %-k. This holds for every k E N, q.e.d.

A sequence { f,> c (5 is weakly convergent if, for every g E G', the sequence (9, fi) converges.

THEOREM 4.3a. Let {h} c '3 be weakly convergent. Then lim, /If, - f I l k = 0 for every k E N, wheref( z ) = lim, f i ( z) .16

Proof: (a) For every k E N , the norms l]Allk are bounded. In fact, on the Hilbert space BPk the sequence L,(g) = (f, , g) converges, hence the norms

I(Li(llk are bounded, and Ilf,l(, (b) By hypothesis, fi(z) = (e, ,k) converges for every t. 13y Lemma 2, Section 2.2, the assertion follows from (a) and (b).

- Z / C - ~ ( (Li ( l Ik (see Section 3.12).

4.4. Linear operators on (E and on (3'. Let X be a linear mapping of @ into itself. Xis continuous-in the topology of @-if to every Bk, l : ( lJ lk 1, k E N)

there exists a closed ball Ek,,Tk: ( I f Ik' 5 7 3 such that XBb,,rk c Bk,l . Equivalently, X is continuous if for every k E N there exists a non-negative integer k' and a positive constant clk such that IXf I l C 5 uk If 1,. , clk =

Lastly, X is continuous if and only if it maps every null sequence in (5 into a null sequence. (The proof is standard and may be omitted.)

-

- -

.

OPERATORS ON (5'. A linear operator Y on (2' will be called continuous if it maps every null sequence in (5' into a null sequence.

THEOREM 4.4. A linear mapping Y : ((5' + (5') is continuous if and only if, f o r evep k E N, there exists a k" E N and a positive bk such that I Ygl.+" & 1g1-k f o r g E EPk.

Thus weak convergence in Cf is equivalent to s t ron~ convergence in Cf (ix. , convergence in norm as defined in Section 2.3).


Proof: Sir$Fcienq of the conditions. Let {g,} c 0.' be a null sequence so that lirn, lgi lpk = 0 for some k E N. Then IYgll-k- /& Igll-r, i.e., ( Y g , ) is also a null sequence, q.e.d.

.i..eressity of the condition. (Proof by contradiction.) Suppose that Y is continuous, but that the condition fails to hold for somek. Then, for every j , 1 YglPi is unbounded on every ball lgl-7c 5 r in EV. Consequently there exists a null sequence {g,} c @-k

such that Igj(-k 5 ( j + 1 )-' and I Ygjl- j > j , j E N. Since Y is continuous, { Ygj} must be a null sequence and 1 Yg,l-, bounded for some i. However, I Yg,l-, 2 j Y s , ~ - ~ > j i f j > i , which establishes the contradiction.

PROPERLS BOUNDED OPERATORS. Most continuous operators Y on 0.' to be discussed in this paper have the property that they map @ (considered a sub- manifold of @') into itself and that this mapping is continuous in the topology of @. I n particular, the following special class of "properly bounded" operators will be encountered.

DEFINITION. A linear mapping Y : (E' -+ 0.') is properly bounded if f o r every real p

(4.1) llygIlp-" s Y p lklIp > g E E p ,

with positive constants yp and a Jixed v.

It is an immediate consequence that linear combinations and products of properly bounded operators are properly bounded.

For a properly bounded Y one finds: 1) Y defines a continuous mapping of @' into @'. 21 I' maps @ into itself, and its restriction to @ is continuous in the topology of E. 3 ) The constant v may be replaced by any constant v' > v without changing the y p .

The above definition may be stated either for the set \ . I p or for the set ll.lJp of n o r m . The values of y p and of v will in general change when one passes from one to the other. By Theorem 3.2, it suffices in all cases to replace v by any v' > v + n.

4.5. Adjoint operators. O n the basis of the results obtained on linear functionals and on operators one may establish the existence of adjoint operators in standard fashion. 1) For every c.L. (= continuous linear) mapping X : @ -+ E there exists a unique c.1. mapping X * : E' -+. @'-the adjoint of X-such that, for all g E E', f E @,

i4.2aj !5, w, = ( X * g , f) * 2) For every c.L. mapping Y : E ' -+ 0.' there exists a unique c.1. mapping Y * : & --f @-the adjoint of 2'-such that, for all g E @',f~ E,

(4.2b) (y.5 f> = (5, Y * f ) .17

l' To establish (+.Zb) one needs the results of Section 4.3.

52 V. BARGMANN

The usual relations are of course valid, such as

( X * ) * = x J (XI+X,)* =x: + X; , (X*X,)* = X t X ; " .

Note the following application. Let Y be a c.1. operator on (5'. If g, is an (This follows analytic functional, so is h, = Yg,, and ah, /dqj = Y(i?g,/aqj).

from the relations (h, , f) = (g, , Y*f) .) With the help of the adjoint operator i t is

easily shown that every X (or Y ) may be expressed as an integral transform. In fact, let h = Xf , and k , = X * e , . By (4.2a), h(w) = {e, , Xf) = ( k , ,f), so that

INTEGRAL REPRESENTATIONS.

(4.3)

- where X(wJ z ) = k,(z) = ( X * e , , e,) = (e, , X e , ) (compare Part I, Section Id). An analogous relation is of course obtained for Y. In particular, if Y = X*, then Y ( w , z ) = X ( 2 , w ) .

PROPERLY BOUNDED OPERATORS. Let Y be a properly bounded operator, and let X be its restriction to (3 (so that X c Y ) . Then the two equations (4.2) define two adjoints such that Y x c X * .

X* is properly bounded.

Proof: Let g E sp, f E (5. Since Yf = Xf, one obtains from (4.2a) and (4.1)

Thus L ( f ) = (X*g, f) is a c.1. fcl. on 5y-pJ and by Theorem 3.5,

i.e., X* is properly bounded, the constant v remaining unchanged. On (5, Y* = X*. For properly bounded Y we shall hereafter identit? the adjoint Y *

with its properly bounded extension X* on E'. (Since every element of (5' is the limit of a sequence in a-see end of Section 4.l---X* is the uniquely defined continuous extension of Y*.)

Thus it becomes meaningful to speak of selfadjoint operators Y. For example, 1 * = 1 (unit operator) or, less trivially, Q: = Q A if A < 1 (see Lemma 1, Section 3.9, and equation (3.23)).

THE OPERATORS Jb . Let Jb = b z, Le.,

( J b g)(z) = (' ' ' )g( ' ) > b E a 3 , .


(4..5a) lJb&-l 5 I4 lglp > l l J b ~ l l p - l 5 lbl llgllp * '1 Iiu5 J , i q properly bounded. Its adjoint is

Jt = b . V = 2 6, d, , rl, = alaz,. j

Iri fact, let h = JE g. Then h ( a ) = (e, , h ) = (J,e, , g) = 2, b,<z,e,, g) = 2, b, a,e/au, (see equation (1.23a)). From (4.4) and (4.5a) one finds

\4.5b) llJ2 &* 5 4\/cp-l lb ! ! !g!lp . For a considerably sharper bound see Section 4.8.

4.6. The operators V, . The operators V, introduced in I, Section 3a, may be regarded as operators on 3 (here g stands for U or c) :

54 V. BARGMANN

i e . , V: = V,-l , V,* = In view of equation (4.3), it suffices to verify (4.8b), (4.9b) for h = e, , f = e, ; (4.8a), (4.9a) follow then from the first equations in (4.6a), (4.6b).

In analogy to the last equation (4.6a) we have

(4.10) VUJ,Vu-.i = J,,' , VUJ,*V,-i = . J Z , b' = U b ,

for the two operators introduced in Section 4.5.

LEMMA 1. Let f E EP (or f E 5". If lim, g, = g, then, for any (T < p,

lim, I V g v f - V g f l , = 0 (or 1% IIV9,f - Vgfll, = 0) .

(Convergence of g, is defined as convergence of the matrix elements of lJv or of the components of c, .)

Proof By Lemma 2, Section 2.2 (or the last paragraph of Section 3.11), it suffices to remark that, for every z , lim, ( Vg, f ) ( z ) = ( V , f ) (z), and that the norms

IVg, f lp (or 1 1 VgVfllp) are bounded (see (4.7a), (4.7b)).

The one-parameter group V,, and its infinitesimal generator. For fixed c and real u, T ,

by (4.6b). For any f E 3 setf(T, z ) = ( V r , f ) ( z ) . It follows from the definition (4.6) that, for fixed z ,

(see Section 4.5). Furthermore, for fixed z,

a - (V, f) = A,(',, f) = ',,(*c f) aT (4.12a)

(Derive (4.1 1) with respect to (T, and set cr = 0.) Thus A, is the infinitesimal generator of the group V,, . We note that A, is a properly bounded operator on (5'. Furthermore, by (4.12) and (4.10),

(4.13a) A: = -Ac ,

(4.13b) V,A,VU-i = nuc I We prove next that (4.12) remains true if interpreted iis convergence in norm.


LEMMA 2. Let f E 3p. Then (i) A, f E Sp--l, (ii) f o r real T # 0,

(4.14) lim IT-^( VT, - 1) f - A, f lla--l = 0 , a < p . 7-0

Proof 1) (i) follows from (4.5a), (4.5b). 2) In view of (4.12a) we have, for fixed z ,

Since A, f E sp-l and V,,, is bounded on %”--l (by (4.7b)) the right-hand side defines an integral on the Hilbert space SP-l (see Section 3.13) and hence also on any p-l, (T < p. By Lemma 1, / I (V,, - 1) A, f Ila-l < E if 17’1 5 1.1 < TO( E ) ,

say, and the assertion (ii) follows.

COROLLARY. The relation

holds in the topology o f i3 ;f f E i3, and in the topology ofi3’ iff E E‘. 4.7. Further relations. In analogy to equation (3.15b) of Part I we intro-

duce the operators

(4.15) 8, = 2-ll2(dj - zi ) , gj = 2-1 (d, + z,) , dj = ,

which are properly bounded on (3’. Set 2/26 = a + ib, a, b E R , . Then

so that a j = ej . a and qj = e, + q, where ei is the real unit vector whose k-th component equals di, . Note the relations

- - - (4.15b) [a, , girl = djY , [q, , qr] = [ a? , a,.] = 0 ,

In (4.15b), [Yl , Y2] denotes the commutator Y,Y, - Y z Y l . Equation (4.15~) follows from A: = -A, .

Applying (4.13b) to real orthogonal transformations 0 one finds

(4.15d) Vo(a * a”) Vo-l = a’ * a” , Vo(b * g ) Vo-l = b‘ . #, Q ’ = Oa, b’ = Ob .

56 V. BARGMANN

For the applications to distribution theory the following is important: Let - f l = a , f , f ; = qjJ Then

(4.16) fl- = 2-112 di f- ) f ; = 2-112 djff , J"(z) = ~ * Z 2 l Z f ( 2 ) *

More generally, if c = a' is real, and g = V, f, h = - 11, f, then

(4.17a)

(4.17b)

g- (z ) = f-(z - a ' ) ,

h - ( 2 ) = a' * Of-(.).

In Section 5 some of the expressions V6g will be needed, where JQ( zj = A (z, q ) , see (2.13), Section 2.10. Specifically,

(4.18a) v, JQ = J,, , 0 real orthogonal,

(4.18b) v-, 8 = e i b , ( ~ - ~ 1 2 ) $ q--a 9 d 2 c = a + ib . Equation (4.18a) follows from the obvious fact that ,4 is invariant urider a simul- taneous orthogonal transformation of z and q, so that A ( z , q) = A ( O z , Op) and A(O-'z, p) = A ( z , Oq). Equation (4.18b) is equivalent to the identity

- Q

(4 q - 4 e - G ' ( Z + C / 2 ) A ( z + ) = e i b ' ( ~ - a / 2 ) A , q

4.8. Improved bounds for J$ . 1) For the norms lJ.llp ,

(4.19) P 2 0 ,

p < O .

Proof: Since V C preserves norms, it follows from (4.10) {hat 6 may be chosen in the Let f = C M,U, E $ P t 1 , and form 6 = (161, 0, * * - , 0).

g = l b l & , ~ l a,dmlu,~ , where m' = (ml - 1, m2 , * . , mn), Thus Then g = J$ f = 161 af/az,. -

llfll;+l = I: qy;; l%I2 2 11g11; 5 PI2 2 I 4 Vf,I-l 1",,,12 . By (3.1999, (mi Y&,+~ 5 ~ p , f f . It follows from (3.19b)) (3.19h) that

14 rfml-l s Iml ?fml+l s rlf)m+: J

Iml Tf?nl-l 5 (1 + lPU2 It4 Tpm,+l 5 (1 + IPD2$!3 9

p 2 0 ,

P < o , which establishes (4.19).

2) For the norm ) . I p , (4.19a) IJ$flp-l 5 2e1'2(%)1p1'2 161 I f I p .

Proof: Let g = J z f = b+Qh 6 # 0. For fixed z,

HILBERT SPACE OF ANALYTIC FLINCTIONS-11 57

Hence i g i z j i i r-l m axl+,. l f (z + d)i. i j ;,,@,,(:’). ;\ fairly straightforward analysis leads then to the estimate (4.19a).

Choose Ibj Y = $e?,(z\, and use l f ( z” ; \ 5

5. Distributions

\\.e utilize the function spaces @ and (3‘ for a survey of some basic operations on distributions and for the proof of some general theorems (such as the representation of a distribution as the derivative of a continuou., function (Section :i. 1 7 t and the n’uclear Theorem).

‘1-here are four parts : 5a (Sections 5.1-5.9) Basic operations (differentiation, multiplication by polynomials, Fourier transform). 5b (Sections 5.10-5.13) Tensor products. 5c (Sections 5.14-5.1 5) Bilinear forms and the Nuclear Theorem. 5d (Sections 5.16-5.17) Inversion of differentiation.

5a. Basic Operations

5.1. Complex conjugation. If y E .44, its complex conjugate i5 given by - Y;(qt = y(q) . For a distribution u we define 5 by

(5.1)

[,which is natural for a regular distribution), and we call u real if fi = u.

Let ~1 stand for y or I ! , and set g = Ay, h = A@. Then g(z) = {p, x,> (see equation (2.16j) and, since j z = xi, h(z) = {@, x z } = {p, xi> = g(i). To the conjugation of p corresponds thus an operation K defined by

- __

for any g E 3. I f g = C u,u,, then Kg = C Ei,u,.

i.e., IKhl,, = Ih/ , and l\Kh[l,, = [[hll, . Note that-in analogy to (5.1)- K is an antitinear involution, I t is obvious that it preserves both sets of norms,

(5.1 b)

whenever (9, f ) is defined. Kh = h-corresponding to a real y or a-if and only if h ( z ) is real for real z .

5.2. Approximation by regular distributions. From the example a t the end of Section 4.1 we deduce

PRoPosinm 5.1. Every distribution u is the limit of a sequence of regular distributions u j , and the functions u,(qj may be chosen in 9.

58 V. BARGMANN

Proof Let g = Av = X {v, ym}um . Two sequences (hi} c (3 which converge to g are listed in Section 4.1. The corresponding functions v, = A-'hj belong to 9' and are constructed as follows: 1) v l = A-'g,, i.e.,

2) u A = A-lQ2,g, A < 1, lim I = 1. Now QAg E a,, if I < A' < 1 (see Lemma 3, Section 1.17) and an estimate for Iun(q) l follows from equation (2.12) in Part I (with A' instead of A). By equation (2.14) and equation (3.23),

Thus

where on(A, q, 4') is the function defined in I, equation (1.19~).

EXAMPLE. Choosev = 6, (see Section 2.10). Thenv,(q) = 2 p,(b)pl,(q) and v,(q) = crn(A, q, b ) . If b = 0, then u2,(q) is the Gaussian I m I s ' l

[n( 1 - P)]--n/2 exp (3q2( 1 + A"/( 1 -- A2)} .

5.3. Operations on 9'. In Sections 5.4-5.8 some basic operations on distributions will be surveyed. I t will turn out that they correspond to properly bounded operators on (3' (Section 4.4-4.5), and we draw a few general conclusions from this fact.

Let g = Av, v E Y', and let Y be a properly bounded operator. To the mapping g, = Yg corresponds then a mapping

(5.2) v1 = Pv, F = A-lYA,

which may also be uniquely defined by

(5.2a) {Fv, y } = {v, P*y}, P* = A-lY*A.

Once the connection with a properly bounded operator has been established one may infer: 1) The mapping v1 = f v is continuous in the topology of 9'. 2) The mapping y1 = P*y is continuous in the topology of 9'. 3) Let 8 be the restriction of P to 9. Then a m a p s Y into itself and is continuous in the topology of 9'; in addition, P is the unique continuous extension of 8 to 9'. (If v = lim, vj , vj E 9',-as in Proposition 5.1-then Pv = limj xvj .)


I t has been traditional in distribution theory to define a mapping pu by an equation of the type (5.2a) where P* is some operation of classical analysis. Generally, also 3 is an operation of classical analysis. (The best known example is differentiation.)

The new feature introduced by the mapping A Y ‘ = @‘ is that in many cases Y itself remains an operation of classical analysis on (5.’.

5.4. The operators Ta,b . In Sections 5.4-5.8 we shall study transforms of the operators Vintroduced in Section 4.6. Set P = A-lVA. Since Vis a properly bounded operator (i-e., defined on (5 and on E‘), the transformed operator p is, accordingly, defined on Y and on Y’, and on 9 its action is described by operations of classical analysis.

begin with p, = A-lV,A and set

(5.3) Ta,b = P, , c = 2-’/2(a + ib) , a, b E Rn .

‘To determine Ta,b on Y let f = Ay, and y1 = T a , b ~ . Then, by (4.9b) and (4.18bj,

yl(q) = (6, , V , f ) = (VJU ,f) = e-ib-(q-a’2) (8q-a ,f) >

Since Vy = V-, , we have, for a distribution v ,

‘This equation may be taken as a definition of Ta,b on Y’.

THE TRANSLATIONS Ta . If c is real we set Ta,o = T, , Thus, if y E 9,

( 5 . 3 ~ ) Y l ( d = Y ( Q - a ) , PI = TaY 7

and we obtain the “translation” of the function y by a. Let p; stand for either y or u, and let v1 = Taq. If h = AT, h, = AT,, then,

by equation (4.17),

(5.3d) h;(z) = h-(z - a ’ ) , a’ = aldT.

As an im- 5.5 Differentiation and multiplication by monomials. mediate consequence of Lemma 2, Section 4.6, we have

PROPOSITION 5.2. If y stands f o r either y (E 9) or v (E 9’), then f a r any a , b and real T (# 0 )

(5.4)

60 V. BARGMANN

where respectively .

Thus, by (5.3a),

= A-lA_,A. For y or v the limit is taken in the topolopy of Y or Y',

Let y E Y and y 1 = A-,y. Equation (5.4) implies point-wise convergence.

(5.4a) Y l ( d = a - a Y ( d + i (b ' d Y ( 4 ) 2 y1 = L Y >

(5.4b) hJ, y> = {v, b l > V E Y .

where a a stands for C a3 a3 . Since A:, = A,, by (4.13a),

For a distribution v we also write

(5.4c)

i.e., this equation is a definition of its right-hand side, while I T - ~ V in turn is defined by (5.4.b). Now n-, is linear in the real vectors a, b. Thus we obtain

15.5)

L C v = a . aU + i ( b . q)v ,

(a av, y~ = +, a - aw} , f a l v , WI = +, a3y1 ,

( ( b ' Y ) V , Y > = 64 ( b * q)wI 9 {qP, Y > = { v , 43w) *

If b = 0, Proposition 5.2 yields what L. Schwartz calls the "topological" definition of the derivative of a distribution, namely,

lim ~ - l ( T-Ta - 1 ) v = a - av I

r-ro

Comparison of (5 .4~) with (4.15a) shows that

A 8,A-l = zj , A q3A-l = q3 .

Using (4.16) we obtain by iteration, for p = v or y,

(A-v) > (5.6a) A-(a["Ip) = 2-1"1/2 d [ m l

(5.6b) A+(g["l,p) = 2-1"//2 d["I(A+p)

(5.6a) contains the assertion that a3(akp) = a,(a,y).

(1.12) and ( l . l l a ) , ) (For y and for certain regular distributions, (5.6a), (5.6b) correspond to

5.6. Examples. 1. Derivatives o f the &function. By (2.1 3), & ( z ) = n-n/4e-22. Hence

2 1 m I / z ~ - ( & m l s ) = d [ m l f i = ( - ~ ) [ ~ I ~ - W H ~ ~ - Z ' ,

where H, is an Hermite polynomial. Conversely, let g ( z ) = p ( ~ ) e - ~ ' / 2 or g+(z) = p ( z ) , where p is a polynomial of degree 1. Then p may be expressed as a


linear combination of Hermite polynomials (see (A.21)), and u = A-19 = ;I: y,lta["16 with suitable constants y m .

InLISz

If a E R n , then 6, = T,6. Hence by (5.3d),

A(8"16,) = (-1)1"12-'"'/2H,(z - a ' ) A ( z , a ) , a' = aid?. 2 . In order that u = 2 y m arm's, it is necessary and sufficient that

q[m'lz = 0 whenever Im'l = I + 1. (In fact by (5.6b) this condition amounts to d[m']g+ = 0 whenever Im'l = 1 + 1 (g+ = A+u) and is satisfied if and only if gf is a polynoniial of order j 1.) If n = 1 one obtains the criterion qZ% = 0.

3. By 2, Section 1.13, A-(q["I) = 2-'"'/2(4n)n/4~,(z). Every polynomial of order 5 I is a linear combination of the polynomials H , with [ml 5 1 (see (A.21a)). Thus g = A-v is a polynomial of order 1 if and only if v is a polynomial of order 1. Consequently, by equation (5.6a), v is a polynomial of order 1 if and only if a[% = 0 whenever (m( = 1 + 1. 4. The equation q dvldq = Iv ( n = 1, 1 an arbitrary complex constant). See [I I],

Volume I, p. 132, equation (V, 6 : 17). This equation has two linearly independent solutions, and Schwartz remarks that the order of the equation no longer determines the maximal number of independent solutions. In this case the classical relation is restored by the mapping g = A v , which leads to the second order equation

Iml 61

- d2.Y 2q ag = (d + . q d - .lg = - - ( 2 2 + qg = 2ng. d 2

It must be proved, of course, that the two independent solutions belong to (5' (this is discussed in Section 8.8).

5.7. The operators Po = A-I VoA. For a real orthogonal transformation 0,

Yl(q) = Y(0+3) > w 1 = P O W , W E Y .

This follows from equation (4.18a), since we have

Yl(4) = (8, > V o f ) = (Vo-18, ,f) = ( L l * ,fi , f = A y .

For a distribution u, ( P o u , y } = {a, ?,-,y}. The case 0 = -1, though trivial, is important because P-, serves to define

even and odd functions (or distributions), i.e., Q is even or odd according as = Q: or P-,cg = -q (for p = u or 111).

5.8. The Fourier transform. Let ly E 9', and let pl = B y be its Fourier transform. It has been shown in Section 3b of Part I that F = P l 0 , where Uo = i . 1.

62 V. BARGMANN

We prefer to derive the relevant facts directly. Setting t b ( q ) = e-Zb‘Q, b E R , we have

By 3, Section 1.13, ( 2 ~ r ) - ~ / ~ A t , = Vc;& , so that

d P ) = ( q 4 ,f) = (8” 3 GJ-) Y

Hence S y = A-lVu0 f = A-lVuU A y = Pu0y as asserted.

f = A y .

For distributions we maintain the notation 9 = PL,o , Thus

{ F u , y } = {u, F - l y } and

It follows that 9 is a continuous linear mapping of Y into itself (and of Y’ into itself).

{Fu, F y } = {u , y } .

We also set $ = VL,, , so that, for every g E 3, (5.7) (9&2) = g( - i z ) .

(5.7a) g’(2) =f-(-i.), g - ( 2 ) = f + ( - - i t ) . If g = g J f ~ 3, then

From (4.6a) and (4.13b) we obtain

Equation (5.8~) implies that 9-l = Fp- l , which in turn yields the standard inversion formula for y = F-lp, p E 9,

(27r)“”%+9(g) =/e-’@”p(-p) dnp = e i Q ’ ” v ( p ) d * p . s We draw further conclusions from (5.8). 1 ) By (5.8d), 39, = P o 9 , which

expresses the orthogonal invariance of the Fourier transform. 2 ) Let c = 2-1/2(a + ib), ic = 2- l I2 ( -b + ia). We deduce from (5.8e), (5.8f)-using (5.3) and (5.4c)-that

(5.9a) FTa,aF-l = T-b ,a ,

(5.9b) 9~~F-l = i a,. , F a j F - l = ig,; . l8 This definition of the Fourier transform suits the formalism of this paper (see equation (5 .7 ) ) .

It differs from the definition in [I 11 and from that in [6].


5.9. Examples.

1. 9 - 1 = (2n)n:2 6 , F d = (2*)-"/21 .

Here 1 is the regular distribution v ( q ) = 1. By 1, Section 1.13,

~1 = pn) n i ~ ~ - n / 4 ~ z ~ l Z

hence g ( A 1 ) = ( 2 ~ ) ~ / ~ 8 , $8 = ( 2 ~ ) - " / ~ A l . 2. 3 T q l m I = ii"'(2n)"!2 p 1 8 , $T(arml 6) = ;'"l(2n)-"/2q[ml .

Apply (5.9b) to example 1.

3. Ftb = (2n)n'2 6, > 9 6 , = ( 2 ? 7 ) - n ' 2 t - b , t b ( q ) = e*D'Q.

Let t, = A t , . By 3, Section 1.13, t b ( z ) = ( 2 ~ ) " / ~ A ( i z , 6 ) = ( 2 ~ ) ~ / 2 A ( - i z , -6), and &,(z) = A ( z , b) .

5b. Tensor Products

5.10. Tensor products on get and @Ai. 1) Let 3,, , 3,, be the spaces of holomorphic functions in n1 and n2 variables with elements f ( z ' ) and g(z") , 2' E C "1 , z" E C,, . Along with 3n, we consider the space 3 = 3,, n = n, + n2 , of holomorphic functions f ( 2 ) = f ( d , z") where

t It 2 = (z', 2") = (.I, . * * , z n l , z1 , . ' . , z;%) .

In 3,, and 3 we construct Cfiz , (E,Lz , , i = 1, 2, and (50, (5, (5' by means of the functions O i l ( z l ) , O;l(z"), O;l(z) and the corresponding norms l . l p , I and ] . I p . The brackets in 3,, and 3 will be denoted by (,)i and (,), respectively.

AN INEQUALITY. Since (zI2 = 12'12 + 1z"I2, we have e;(z) 5 O~(z ' )0 ; ( z f1 ) 5 e;,(z) if u 5 0 (see (A.5)). Hence,

Letf(z) = f (d, 2") E 3. For fixed z1 we define the element f , . of 3,, by f,, ( 2 " ) = f ( z ' , z"). Then by (5.10),

(5.1 1) Ifd1,.2 5 I f l e p e y z ' ) > p 2 0 .

TENSOR PRODUCT. Iff, E 3,, , i = 1 ,2 , their tensor productf =fi g fz (E 3) is defined as

f ( 4 = (fi @fi)(z) =f1(z'lfi(z") *

I t follows that f+ = f; 0 f : , and f - = f; 0 f; .

64 V. BARGMANN

Important examples are:

r rr Here rn = (rn‘, m”) = ( m i , * * * , mnl , m, , *

(5.11a)

* , rnlz). Letfi E Eg: and p = min ( p l , p,). Thenfi E (E;, and by (5.10)

I f 1 @fZl$ 6 /flIp,l ’ If2lp.2 J f1 of, E Q? .

It follows that f i 0 f i E E i f f , E Enz , and that g , 0 g, E t!? i f g , E EL, .

that f l E (%El , f2 E Eiz .

(5.12)

Conversely, let f, E 3ni and f = fl 0 f2 E Ep. Iff f. 0 it is easily shown

2) Since d,un(z ) = d,unl(z’) d,unI(z”),

(91 @ g, , f l of,> = ( S l , f l > l - (9, ,&>, >

whenever both factors on the right-hand side are defined. In fact,

PROPOSITION 5.3. A continuous Linear functional (g, f) on E is uniquely deJined by its values for all f Ofthe form fl 0 f, ,f, E Eni .

- Proof: Note that g(a) = (g, e,) = (g, e,, 0 e,,,).

As a consequence we obtain from (5.12)

PROPOSITION 5.4. Let gi E Ek, . There exists a unique element g E E‘ which

sati?bes (9, f1 of,> = ( g l > f1)l * (92 J fz), for a L L f , > g = g 1 0 g 2 .

5.11. Tensor products on %, and <l. We consider functions y(q’) , y(q”) and y(q) , 4’ E Rnl 3 4‘’ E RnZ 7 q E Rn > n = n 1 + n 2 > where

I I’ q = (q’, 4”) = (41 , * * 3 q,, 3 q1 > - * * q:J .

In particular we are concerned with the corresponding function spaces Ynf and 9 defined by the norms ] I * ) \ $ % and I]./\,”, respectively. (For distributions we write {,}% and {,}.)

Let y(q) = y(q‘, q”) E 9. As before we define, for fixed q’, the function yot(q“) = y(q’, 4”). Then yQt E ynz and

(5.13) Ilw,4$z s rlylr:.


and the inequality (5.13) follows.

TENSOR PRODUCT. For y 1 E %, , i = 1 ,2 , the tensor product y = y1 Q yz is defined by

Y(4) = Yl(Q')YZtOz(Q~') .

It is easily seen that y E 9, more precisely, IIyl O yzlif 5 IIyljl:l . jly2jjp2 .

CONNECTION WITH Ent AND E. With the help of the kernels Anl(z ' , q') , A n 2 ( z " , y"), and A ( z , q ) we define the mappings A, , A , , A , respectively, so that A , qt, = E n , a n d A Y = E. By definition (1.7), A p , q ) = Anl(z ' , 4') .Anz(z", 4"). Thus iff, = A , y I , yz E Yn, ,

(5.14) A(Y1 O ~ 2 ) = A i ~ 1 0 Azy2 =fi Oh * 'rENSOR PRODUCT OF DISTRIBUTIONS. Let ui E YiLg . We define:

or, equivalently, A(u,

obtains from (5.14), (5.15)

u2) = A l v l 0 Azu2 = g, 0 g2. Translating Propositions 5.3 and 5.4 into the language of distributions one

PROPOSITION 5.3a. A distribution (u , y } on 9' is uniquely determined by its values for all y Ofthe form y1 8 y z , y , E Yn, .

PROPOSITION 5.4a. Let u, E Yn'* . There exists a unique element u E 9' which satisfies { u , v1 3 yz> = {u, , yl}l . (v2 , y2) , f o r all y z E Yn, , namely, v = u1 @ uz .

5.12. Standard form of the tensor product. I t remains to express L J ~ 3 uz more directly in terms of v1 and v2 , i.e., to eliminate the operators A and A, from its definition (5.1 5), and thereby to recover Schwartz' original definition. We proceed as follows:

1) Let g, = A , u , , h an arbitrary element of @il, and f = A y E E. Then

(5.16) (h 0 g, ,f) = m 1 >

(5.1 Ga)

66 V. BARGMANN

It follows easily from (5.1 1) t h a t j is holomorphic and belongs to all @El , so that j E . For h = gl,

{UI 0 u 2 , y> = (Sl 0 g, ,f) = (91 ,j>l = {UI > A;lj), .

2) We assert that, in close analogy to (5.16a), A;? is given by rp(q') =

(a) L,,(y) = q ( q ' ) is a linear functional on 9. Assuming I ( v z , 1}21 j c l l l , l [ 2 2 , we find from (5.13) that IL,.(y)l 5 c lly,.@z c llyll$. This shows that L , is a distribution on 9. (b) If y = y1 0 yz , then yqj(q") = yl(q ' )yz(qN), and hence

(02 > Wd>2 - Proof:

LQ'(yl yz) = y1(q'){'Z > Y2)a = {6Q' J y1}1 * ('2 > yZ}Z .

I t follows from Proposition 5.4a that L , = 6,. 0 v2 . 'Thus

and, by (2.14), y = A-ti as asserted. We have proved

PROPOSITION 5.4b. Let ui E Yk5, and y E 9. Then

(Apart from the notation this coincides with Schwartz' definition ([ 111, Volume I, p. log).) It is obvious that the roles of u1 and uz may be interchanged.

5.13. Additional remarks. We list a few general properties of the tensor product which are all simple consequences of its definition and of the estimate (5.1 la). (Most of these will be formulated for the spaces @ and E', the translation into the language of Y and 9' being straightforward.)

(a) Convergence. Letfi ,f E 3,i and hi = X I -f, . Then

f[ Of; -f1 Of2 =f1 O h, + h1 Of2 + hi O h,.

Consider two sequences {f,,i} c (Jig;, i = 1, 2, and set p = min (pl , p z ) . In virtue of (5.1 la) we infer from the last equation: if


As a result we have

PROPOSITION 5.5.

, z = 1 , 2, then limv (gv,l @ gv,2) = g, 0 g2 in the topologV of(5’.

(b) Operations on tensor products.

If limy j i , t = f , in the topology of En*, i = 1, 2 , then limv ( f v , , &.fV,,) = fi 8 f2 in the topology .f (5. If lim, gv,i = g, in the lopology of

(Here f, E 3,, .) We mention only the following. 1) Let c = (c’, c”) E C , . Then V,( f, 0 fi) = Vc, f, 8 VC,, fi , and ilc(J, (2 f 2 ) = A,. f l 0 f2 +fl E ACT, f 2 . 2) (sf, O h ) = sIJl 0 F2,fz . Here, .< is defined on 3,, by equation (5.7). Thus the Fourier transform of the tensor product is the tensor product of the Fourier transforms.

(c) A distribution u E 9‘ is independent of q’ if Tau = u for any a = (a‘, 0 ) . For this definition we have the following criteria.

PROPOSITION 5.6. Each of the following conditions is equivalent to the independence of the distribution v of q ’ : (i) a,v = 0, 1 2 j 5 n, , (ii) u = 1 0 v2 , where u2 E xi .

Let g = Au. By definition, v is independent ofq‘ ifg-(z’ - b‘, z“) =

<<-(z’, z“) for any b‘ = a1/d2 E Rnl . This is equivalent to: (i) d?g- = 0, 1 sj nl, (ii) g-(z‘, 2‘) = yg;(z”) = A;1 8 g; for some g2 E @k2 (with y = :47~)~1/4 , see I , Section 1.13) .

Proof

5c. Bilinear Forms and the Nuclear Theorem

5.14. Bilinear forms. We keep the notation used in the discussion of tensor products. The bilinear forms B ( f l , f2), f, E @,&, to be considered are separately continuous in fl , f2 , i.e., for fixed fi , B( fi , f 2 ) is a c.l.fC1. on Gn1 , and for fixed f, it is a c.1. fcl. on E n 2 . (A corresponding definition applies to the bilinear form b ! y l , y2) = B(A,yl , A2y2) on SP,, x 9&.)

A generalization of the proposition stated in Section 4.1 implies that for any ruch bilinear form B there exist two non-negative integers k , and a positive constant y such that IB(f,, fi)/ 5 y ljJkl,l ]fi lk2,2 or, equivalently,

(5.17) IWl?fi)l 5 Y Ifilk,, Ifilk,Z 5 k = max ( k , , k,) , for any f, E En, . (For a simple and elegant proofsee [7], Chapter I, Section 1.2.) The inequality (5.17) in turn implies that B is jointly continuous in fl , f2-in the standard product topology on x En2-as follows from the identity

We shall therefore simply speak of continuous bilinear forms.

68 V. BARGMANN

LEMMA 1. Let B, and B, be two continuous bilinear forms on enl x en, such that B2(e,, , eae) = Bl(ea. , eae) for all a’, a”. Then B , = B, .

Proof: Set B = B, - B, . Then B(ear , em,,) = 0, and bylinearityB(s, ) s2) = 0 for all s, E ~ ~ ( 2 3 % ) . (Here Bl(B2) is the set of all ear(ea8,).) Fix fl , fi . By the Approximation Theorem 2.4 there exist two sequences {sV,,} c sp(%,) such that, for all k EN, lim, Isv,% - f i l r , , = 0. Hence, by (5.17) and (5.17a), B ( f , ) f i ) =

lim, B(s,,, , s,,~) = 0 for all fi ) fz , q.e.d.

EXAMPLE, Let ,D E G-l. Then

(5.18) Wfl , f z ) = (&.A @ f z > 3 f, E en, )

I t suffices to observe that I k , f l 0h)l 5 P 191-2 I f1 Ofilk 5 P lgl-2 lfllk,l IfZlk,2 )

- is a continuous bilinear form on Eel x (En, , and g(a) = B ( e , , ear,) if a = (a’, a”).

where k = 2n + I + 1, and /? =j!2n-l,2n (see (1.22a) and (5.11a)).

type (5.18). We formulate it for b ( y , ) y z ) . The Nuclear Theorem asserts that every continuous bilinear form is of the

NUCLEAR THEOREM. Every continuous bilinear form b(y, , y z ) on yn, x q2

5.15. Proof of the Nuclear Theorem. Since (5 has been proved to be a nuclear space (Theorem 3.4), one may apply the general theory of nuclear spaces. We prefer a more elementary proof quite similar to the proof of Theorem 2.5 on linear functionals. As a result, the proof depends only on Section 2 (through Section 2.8) and a few simple facts on tensor products, mainly the example in Section 5.14. We proceed as follows.

(a) Let B ( f l , fi) be a continuous bilinear form which satisfies the inequality (5.17). Define

a = (a‘, a”) E C n .

may be expressed as {u, y1 0 y2) with a uniquely determined element u E 9’.

g(a) = g(a’, a’’) = B(e, , cum) , BY (2.3),

(5.19) Ig(a)I 2 Y lea, Ik.1 leu‘ lk,2 5 yle:(a’)ek(a‘’) 5 ~le,lk(a)

with a positive constant y, . (b) Analyticity of g. For fixed a, we analyze the difference g(a + b ) - g(a),

where b = (b’, 6”) and lbI2 = 16’l2 + lb”I2 2 1. Set ea,+b, = 2 t p ) ) and ea”.ib” = 2 v=o 2 t p ) ) where, by (2.5),

2

v = l

t y = $2) t:) = h l f k , b’,a‘ 3 $0) = 1 ea , ,

t p ) = ear, > $ 2 ) - (2) - rbr,,au . t:) = h ( l ) b”,o” >


By linearity,

r i a + b ) = IF;; B,, = 5 tp)), Po0 = g ( 4 . I r . V

In view of (5.17), lBpvl =( y Jt:”’J,,, . By (2.5a))

l t i“)Ik, l = o ( l b ’ l ’ l ) = o ( l b l p ) >

and = 0(~6”~”) = O(lb]’) ,

so that lpp, l = O(lbl”+’)). As a result,

g ( a + 6 ) = d a ) + 4 b ) + o(lblz) , = Blo + Bol . Since h$,)*. and h$la. (and thus Bl0 and Bol) are antilinear in b’ and b”, respectively, l ( b ) is linear in b. It follows thatg(a) is holomorphic, and, by (5.19), g E EPzk c E‘.

(ci By (5.18)) Bo( f l ,fi) = (g,fl ofi) is a continuous bilinear form, and

B(e,,, , can) = B,(e,, , ea,,) = g(a ) . We conclude from Lemma 1, Section 5.14, -

that B V , ,f,j = BOK ,fJ = {g,J1 O f A q.e.d.

5d. Inversion of Differentiation

5.16. The equation a jw = u j . Consider a distribution w E Y’, and let L’, = ajw, 1 sj 5 n. Then a k v j = ajvk . Conversely, we have

PROPOSITION 5 . 7 . Let u j ) 1 5 j 5 t ( t 5 n) be distributions on 9’ which satisfy There exists a distribution the integrability conditions a k v j = a,v, , 1 =( k , j 5 t.

te‘ (E 9‘) sltch that a,w = uj .

Proof Let gj = Auj ( E ( % I ) . By hypothesis, dkg; = djg; ) 1 _I k , j 2 t , and our task is to find h (E (-5’) such that djh- = d?g; (equation (5.6a)). Then w, = A-lh solves the equations ajwo = u j .

,dssume gj E (%-pj, pj 2 0, set p = maxi pi , yi = Igjl--pj , and y =

The function h- is constructed in standard fashion by setting, for real T ,

2 y; . ( j :

Then /&(z)I 5 yjf$(z)eVy2.

t k ( z ) = 4 2 U ( T , Z) d 7 , U ( T , 2 ) = I Z,&[TZ] ;* -l k=1

h- is obviously holomorphic, and - since zk yk (zkl

Thus h E W-l, and lh\-p-l 5 d 2 y . y Iz I , 1 2 1 ( ~ , .)I 5 yei+l(z)ey2.

* Here g;[~z] stands for ~ F ( T z ~ , . ‘ . , 7 z t , z ~ + ~ , . . ’ , z e ) .

70 V. BARGMANN

Thus djh-(z) = fig;(z) as required. All other solutions of ajw = u j are of course obtained by adding to w o = A-lh

any distribution w1 independent of q j , 1 sj 2 t , (see Proposition 5.6) or w1 = const. if t = n.

5.17. Distributions as derivatives of continuous functions. Schwartz' celebrated theorem that every tempered distribution is the derivative of a continuous function of polynomial growth may also be obtained by a fairly straightforward construction. The result is as follows.

PROPOSITION 5.8. Let u E 9', and A u EE-', 1 EN. There exists a regular distribution v o such that u = arf a t . - ak,v0, k = 1 + 2, v o ( q ) is continuous, and Iuo(q)l 6 x i , , Jg1-$ Ti 8:L+2(qj) with a constant xZ,, depending only on 1 and n.

j

Proof: Let g = Au, and mo = (k , k , * , k ) . By equation (5.6a), we have to find h- such that dcmolh- = Znk12g- (and to set u o = A-lh). A holomorphic solution of this equation is given by the integral

with real T~ (analogous to the expression for r k ( z ) in Section 1.10). To estimate lh(z)I we note that Ig-(z)l Igl-Lf3;(z)P'2 and zj T;$ 5 2, ~ ~ ( 1 +y,"). Thus

1,g- (7121, . - , T , z , ) I 5 Igl-,e;(z) Ti e r j ( l + 4 ) . j

Furthermore, lzilk f3i(zj) and @(z) 5 e!(zj), so that, by (1.15) and (l.l5a), i


Hence h satisfies the hypothesis of Theorem 1.4a. By Section 2.11, v, = A-lh = Wh is a continuous function, and byTheorem 1.4a, Ivo(q)l a z , n lgl-z J-J O:z+2(qj) , where a!., = [e . ( 4 ~ ) ' / ~ ( 2 ~ 1 ~ ~ ~ ) z+tj&+2,1]n. 5

COROLLARY. I f n = 1, u E SP,, and Au E E-', then any solution u ofd1+2vldgz+2 = u is a continuous function, and ]v(q)/ 5 y8,OZ+,(q) for some positive constant y .

Proof: Since dz+2(v - vo)/dgz++" = 0, v differs from v o by a polynomial of order 5 1 + 1.

If n > 1 the conclusion of the corollary no longer holds, i.e., there are solutions u of 8 m o k = u which are not even regular distributions. The particular construction which yields u, assumes therefore a greater significance, and it is worthwhile to characterize u, more directly.

If g-(r ) = ~ m / ? m z m / [ r n ! ] , then, by construction, h-+) = 2kn12 zm @,z["+"ol/[(m + mo)!]. Thusifweseth(z) = ezs12h-(z) = X um,um,(z),

then a,, = (h , urn,) = 0 unless m' 2 mO, and this condition defines a unique solution of d[mOlh- = 2k42p-- . The function v, of the proposition is therefore determined by:

This may be done as follows.

-

Let, for example, u = 6. Then I = 0 (Section 2.10), and one obtains

6. Test Functions and Distributions of Compact Support

Both in 9' and in Schwartz' space 9 distributions of compact support play a significant role. In this section we consider the images-under the mapping A-of both test functions and distributions whose support is contained in a compact set K.

For a convex K necessary and sufficient conditions of the Payley-Wiener type are derived which Ap, must satisfy in order that the support of 'p be contained in K. (Here p, E Y or 'p E Y'.) The methods used were suggested by the work of Plancherel and P6lya [lo].

6.1. Preliminary remarks on convex sets. Let K be a compact convex set in R n contained in a ball IqI 5 c. (In what follows K is keptfixed.) It will be described in terms of its support function ~ ( s ) : for every vector s E R ) M(S) = supneK (s - q) . Note that

(6. * ) la(s)l 2 c Is[ , a ( h ) = 1a(s) if 1 5 0 .

A point p belongs to K if and only if s * q ~ ( s ) for every s.

72 V. BARGMANN

\Ye mention two special sets: (a) KO, the set which contains only the point q = 0. Here ~ ( s ) = 0 for all s. (b) 'The closed ball B, : (1q1 2 Y ) , with EU) = r Is[.

Along with K we consider the following additional compact convex sets: (i) a + K (translation by the vector a ) , consisting ofall points a + q ( q E K ) with support function p ( s ) = ~ ( s ) + a * s. (ii) Kc , the set of all pointb whose distance from K is s E , with support function

(6.1 a) cI,(s) = 4 s ) + & 14 .

6.2. The mapping f = Ay. Let y be a continuous function which vanishes outside K (i.e., supp y c K),Ig and letff = A+y. By ( l . l l a ) , for every multi- index m ,

d["'f+(z) = 21"''2 q["'A+(z, q)y(q) d"g . s, O n K , qr"l I - clrn1, and

I- -

Tn/4 q ) l 5 e - Q 2 / 2 e v 2 X 4 1 < - e -@2/3e t ' 2dZ) - Setting

,- (6.2 ) c('(s) = 4 2 a(.) , c' = c v 2 ,

(6.2a) Idc"'lf+(z)I _I /3c'lmlea'(r) , Y (4 ) I d"q.

we obtain for every m

In addition we have

LEMMA 1. Suppose that y E 9" and supp y c K (.YO that y1 E Y E merely means y EC' ) . T h e n f = Ay E ez, and If+(z)l _I 2'12 I f / , iYt(z)ea ' (") .

Proof: Sincefe (5' (Theorem 1.2a), we have in addition to (6.2a)

(6.2b) If+(z)l 2 l f l l eo,(z)rr2.

We apply the standard decomposition described in Section 1.2, i.e., we set z = (a + ib ( 5 = 6 + iq E C+), and ~ ( 5 ) =f+((a + ib). By (6.1), ~ ' ( x ) = (mi = cc'(a)), so that (6.2a) reduces to ly(c)1 2 PeQ'S, form = 0. Set next

(6.3) Y o ( < ) = ( A + 5)"-"0"Y(l) '

From (6.2a), (6.2b) we obtain: 1) l yo (<) l 5 11 + < I c . 2 ) lyo(iq)l 5 2 z / 2 l f l l (where (1.2) is used). It follows from the PhragmCn-Lindeliif theorem that

lS "supp y" = support of y.


1;1,,{<)1 is dominated by its bound on the imaginary axis, i.e., lye(()[ 5 2'12

in , Solving 16.3) for y ( <) and using ( 1 . 2 ) we find that

/ y ( ( ) l I - 2 l i 2 ~ f i , ( A z + 1c12)-i~2eao't ,

which is the desired inequality. If y E Y, thenfE Q, and If+(.) I 5 2z/2 If1 ,f3!,(z)ea'(r) for every I E N.

6.3. The mapping y = A-y. The following is a certain converse of the last lemma.

LEMMA 1. Let f ~ 3 , and If'(z)I 5 /W,,(z) ea"') for some p > n. Then y = Wf is continuoils, and supp y c K.

Proof: Since la'(x)l 5 c' 1x1, f E QO, and it follows from Theorems 1.5 and 1.6 that p is continuous and bounded, and that f = Ay. For fixed x ,

f-(. + t Y j = n-n/4 J - e ~ 2 ' 2 ~ a ( e - - n 2 / 2 e t / 2 x . q Y ( d ) d"q )

i.e.,f' ( Y + 11') ir essentially2'' the Fourier transform of the continuous and square integrable function yl(q) = e-q212e\ * L " ~ ( 4 ) . By the inversion formula,

-

(6.4j e - i 2 2 i 2 y ( q ) = e-th x.an-ni4 i 2 T ) - n / ~ J ,- i . \ / iv.o f + ( x + GJ) d"y .

Let qo be a point outside K. It remains to show that y(qo) = 0. There exists a vectors such that s . qo - sI(sj = y > 0. Choose x = [s, [ > 0. By hypothesis, If+k + Y)I ' < = BO" - p ( Y )ev'2:a(s), and it follows from (6.4) that e-qoa/2 ly(qo)l 5 - &-t ;2:1 jPp," 0 for all positive 6. Hence y(qo) = 0, q.e.d.

6.4. Distributions of compact support. The mapping g = Av. Let 7; E Y'. Then supp v c K provided the following condition is satisfied: for any y E 9 which vanishes in an open set containing K, (v , y > = 0.

\Ye shall prove analogues of the two lemmas of Sections 6.2 and 6.3.

PROPOSITION 6.1. Let v E Y', and supp u c K. If g = Av E W, then l p ( z ) l i - 2 ! ~ 1 ' 2 Igl-,O~(z)e""r'.

Proof: 1) Let v E ( q ) , 0 < E < 1, be a family of real valued functions in C" such that v , (q j = 1 if q E K,,3 , and v,(q) = 0 if q is outside K, .21 If y belongs to

?" ix., apart from the normalization factor and the factor d2 in the exponential.

This is a standard construction. A suitable choice is ve(q) = p a ( q - q') d"q', where 6,, i, = ~ / 3 and pa is the function defined by L. Schwartz ( [ I l l , Volume I, p. 21).

74 V. BARGMANN

Y so does v,y, and the function (1 - v J y vanishes in an open set containing K. Thus {u, (1 -v , )y} = 0 for all y E Y , or {u, y } = {u, v,y}.

By Proposition 5.8, u may be expressed as the derivative of a continuous function of polynomial growth. For convenience we write u = (-l) lmliYmlw, where acml = arml / [m!] (see Section 1.3), so that {u, q } = {w, Zrm1p} for any p E 9, and 0

{a!, Y } = s W(Q)Y(d d"q. Inserting = vey and applying Leibniz' rule (Section 1.3) one finds

{u, y } = {w, &l(v,y)) = 2 {w, ( % - % J e ) i[%p} . LSm

By construction, w,,& is continuous, and supp wZ,& c K, . 2) Estimate of Igf(z)I. Set g = Au, and h 2 , & = 2-l'i'zAw 1 . E . BY (6.5),

We apply the following inequalities. (a) Fk is a polynomial oforder Ikl 5 [mi-

m. (b) Since supp wZ,& c K, , ld["]h:,(z) I 5 T ~ , ~ ( E ) e a & ' ( X ) (see (6.2a)) it is a multiple of the Hermite polynomial Hk-hence: IFk(z)I 5 oO:(z), p = Iml,

0

for all k with some constants T ~ , ~ . ( E ) . As a result, for every E , 0 < E < 1,

(6.6a)

with a suitable PE , see (6.la), (6.2). Since g E E-P, we also have

lg+(z)I 5 peO:(z)eaL(z) , EL(.) = a ' ( x ) + E' 1x1 , E' = EY'2,

(6.6b) Ig'(4I 5 1g1-,~;(Z)eX2 *

3) We proceed now in close analogy with the proof of Lemma 1 in Section 6.2. Using the standard decomposition z = { a + ib, 5 = 6 + ill, we set y ( 5 ) = g+([a + ib), 5 E C+, and


Here cth = %'(a) , so that or'(x) = x i 6 , and x L ( x j = (xi + d ) E . From now on we assume p 2 0.22 Note that y o and y, are holomorphic on C+ . In view of (1.2j, we find from (6.6a), (6.6b)

(i)

(ii) Ir&(ir)l 2 lgl-, *

b&(5)1 2 P, 11 + 51"-" I

Again we conclude from the PhragmCn-Lindelof theorem that, for every E,

lus(c)l = e-"' Iyo(LJl 5 I&, on C+ . Thus ]yo(5)1 2 Igl-, , and we obtain from

(6.7a) Ig+(.z)I = l y ( ( ) / 2 2 ' ~ 1 / ~ 1g1-P(i12 + 1 5 1 2 ) p / 2 e a i c = 21p1/2 lgI- P P eo(z)ea ' (s) ,

which proves the proposition.

Section 5.4 for the definition of the translation T,) . Remark. Let p E Y or y E 9'. If supp p c K , then supp Tag, c a + K (see

APPLICATION. Distributions concentrated at a point. Suppose supp u = KO (see. Section 6.1), and g = Av. Since c t ' ( x ) = 0, lg+(z)l 5 PO;(z) for some p, and g+

is a polynomial of order 1 p, so that u = 2 y m 8"]6 (see 1, Section 5.6). If 11

is concentrated at the point a, it follows from the preceding remark that u = l rnl51

2 y m 8"16, (since 6, = TJ j . l?nlS:1

6.5. The mapping u = A-lg. The converse of the last proposition may be

Then (i) g EE-P,

stated as follows.

PROPOSITION 6.2. Let g €3, and Ig+(.z)I 5 pe;(z) ea' ( ' ) .

(ii j the support o f the distribution u = A-lg is contained in K.

Proof 1 ) Since Icr'(.x)l 5 c' 1x1 x2 + &", we have @c'g/4. 2) Choose a fixed positive integer k > p + n + 1. Let o(5) = c-l sinh 5 , 5 E C, and set for every F , 0 < E < t ,

T,(z) = o(Az,) , 1 = t ' / k ' / / i , E' = ~2'2 . [jr, I'

It will be shown below that

(a) I~,(z)l 2 ,&'I5/ ,

(b) IT,(Z)l YEO!ik(z)ee'lrl . In addition, lim,40 T,(z) = 1 for every z .

** The proof for negative p requires only trivial changes. (Compare the proof of Lemma 1 in Section 6.2, where ,/I = -l.)

76 V. BARGMANN

Let g,(z) = T,(Z)g(Z) . We assert that lime,, g, = g in the topology of E'. (Equivalently, lim v, = 61, V , = A-l g , , in the topology of Y.) In fact, 1) lim g,(z) = g(z) for every z , 2) the norms lg,l-p are bounded. (Since E' < 1, we obtain from (a) as above lg,l-p 5 /3 exp {$(c ' + l)'}.) This proves the assertion (see Section 4.1).

Lastly consider a function y E 9 which vanishes on an open set 0 containing K . I t must be shown that ( v , y > = 0. For some positive E,, , K,, c 0, hence K, c 0 if E 5 E ~ . By (b), Igz(z)1 2 &ef?On-l(z)e"E("), since ~i(x) = ~ ' ( x ) + E' 1x1. Thus u, = A-lg , is a continuous function, and supp v, c K, (Lemma 1, Section 6.3). If E F ~ ,

Hence {v, y } = lim {v, , y ) = 0, q.e.d. &-+a

Proof of the inequalities (a), (b) : For 5 = [ + iq,

In addition, (iii) lo(<)l = g;21/W'/O;([). (If 2 1, then 2 Illz 2 1 + I l l 2 , and (iii) follows from (i). If 151 < 1, then

For z = ( z , , * * - , zn) set s ( h ) = a ~ ( A z , ) , 0 < il 5 1. Using the in-

equalities cj il 1 . ~ ~ 1 2 Ad; 1x1, and O!(ilzj) 2 O;(Az) 2 lf?,O(z), we conclude

from (ii) and (iii) that

2 > 1 + I l l 2 , and (iii) follows from (ii).)

j

$

Is(ilz)l 5 ,ld%1~1, I s ( A z ) I 2 2 n / V v / ~ I z l / i l f ? ; ( z ) .

Since T, (z ) = ( ~ ( l z ) ) ~ this yields (a), (b), with ye = 2nk'2A-".

7. Periodic Distributions and Their Fourier Expansion

Schwartz obtains the Fourier expansion of a periodic distribution by studying distributions on a torus ([i l l , Volume 11, p. 80). In keeping with the program of this paper we utilize the functionals on (2 instead. If v is a periodic distribution, then g- = A-v is periodic (with real periods) and may be expanded in a Fourier


series. The inverse transformation u = A-lg yields the Fourier expansion of u. The resulting explicit expression for the Fourier coefficients-(7.5)-may be of some interest.

7.1. Preliminary remarks. Let I,, . , I , 23 be n linearly independent vectors in S,, and 9 the lattice generated by them. (9 consists of all vectors

I = 2 v,1, with integral coefficients v, .) In addition we consider the dual

lattice Yd . A vector k belongs to 9, if and only if k . 1 is integral for all 1 €9. Yc< is generated by n independent vectors k , which satisfy the relations k , . I , = bss. , 1 5 s, s‘ n, i.e., k E Yd if and only if k = 2, viks with integral coefficients vt .

For any positive A we define A2 as the lattice generated by the vectors ill, ; its dual is IT’L?~ . Specifically we set 9‘ = 2 - I f 2 2 , and 9; = 2 1 1 2 9 d . We write 1‘ = 2-It21, 1 E 3, and k’ = 2ll2k, k €9, .

By means of 2’ we decompose F? , into cells of the form W , = 2nd + ?V, , 1 E Y , the “base cell” M’, containing all points x = z,3 E,1, , tS real, for which -T 5 E , < P, 1 5 s n. The cells W , are pairwise disjoint, their union is R, , and each one has the volume p = ( 2 ~ ) “ (Al , where A is the determinant of the vectors 1 , .

For any summable functionf,

f d n x = f d”x = I W , J ( x + 2 4 dnx . i.. i E 9 I*, (7.1)

Throughout this section the vectors I , and the lattice 9 are kept fixed, and a complex valued function ~ ( x ) , x E R , , is called “periodic” if it has periods 2nl, so that y ( x - 2 7 4 = y(x) for all 1 E 9. For a periodic q j ,

(7.2a)

(7.2b) P

In (7.2ai it is assumed that q j is summable on bounded sets, and a is any vector in -- ,1 . In (7.2b) we assume that q E c1, thus (b) is an immediate consequence of (a). I_-

For a later application we consider the function

where .Y E P ,1 , x > 0. The series in (7.3) converges uniformly on bounded sets, rs:(x; 9) is continuous and periodic with periods I , and o:( Y ) is finite.

23 Here the subscript distinguishes n vectors and not the n components of one vector.

78 V. BARGMANN

Proof By (A.4), O;.(x - 1) 5 ~ l p ~ O ~ 7 r ( x ) O ; a ’ z ( l ) ~ which proves uniform convergence because c1 O;a’2 ( I ) is evidently convergent. Continuity and periodicity follow immediately; they in turn imply that 0;(2) < co.

7.2. Periodic distributions. A distribution v has periods 2rrl--“is periodic” for short-if TZnlv = v for all 1 E 9 (see Section 5.4).

Assume v periodic, and let g = Av. By (5.3d)) g-(z - 2 ~ 1 ‘ ) = g - ( z ) , I’ E Y’, i.e., g- is periodic with the real periods 2711’. ( A function in 3 will be called “periodic” if it has periods 2711’~) As before we introduce the cells W; (defined by the lattice 9’). Their volume is p’ = 2-”‘2p.

For periodic u, the corresponding g will be classified by the spaces Q: rather than by (51 (see Section 1.14). In fact we have

LEMMA 1. Suppose g E Q’ and g- periodic. (i) If r 2 0, then g E Q: ; (ii) ;f T < 0, then g = 0.

(ii) For fixedy, limizi,m g-(x + ;r) = 0. By periodicity, g- = 0.

In the sequel we shall need the following result.

LEMMA 2. Let g - ( z ) E 3 be periodic, and set for every k E Pd (k’ = 2lI2k)

(7.4)

Then [g-II, is independent ofy.

[g-1, = p’-’Lvi e-ik”zg-(z) d”x z = x + @, ?;&xed.

Proof: The integrand h, = e-zk’‘zg-(z) is 1) periodic, 2) analytic, so that ah,/ayj = iah,/ax,. Thus, by (7.2b),

COROLLARY.

Proof:

I fs‘ E 2: , then [eis”z]k = 68, .

Sety = 0 in (7.4)) and use the standard orthogonality relations.


7.3. Fourier expansion of g-.

LEMMA 1. Let g ED;*, Ig-(z)l s yO:(y) and g- periodic. Then g-(zj =

2 &?k’.z, with constant coeficients BK = [g-]!, , and lj,l k E 9 p d

y 9 i Z ( $ k ’ ) .

Proof: For fixed y , g- (x + 6) is periodic in x and has continuous bounded partial derivatives (with respect to x ) of all orders. According to the elementary theory of Fourier series, g- has the absolutely converging expansion g-(x + ;Y) = zk b,i -y)e2k”x, where

r

By (7.4), b,(y) = e-v’”[g-]k ; thusg-(z) = zk /?ke2K’.zasasserted. For an estimate of /I, sety = -$k’ in (7.4). Then the integrand is bounded by ye-k’2’2ei(&k’), and hence l @ K l 5 y6;z(4k’).

LEMMA 2. Let /?,, k E A?d, be a set of coeficients such that lpkl 5 e i z (+k’ ) . (i) The series 2 k /?,ezk’” converges absolutely and uniformly (on bounded sets) to a periodic holomorphic function g-, and g = eZ2l2g- E Q;. , (ii) The series ezZ1’ zk Bkeik”Z converges in the topology of@‘.

Proof (i) Set ak = e--Y2&etk’.z. By hypothesis,

laKl i y e ; ( g k y q 2 ( y + $k’) 5 yK14;(y)eL!(y + i k ’ ) . Since Oi-;f(y + ik’j are the terms of the series orpl(y; +A?:) (see equation (7.3)) the last inequality establishes absolute and uniform convergence of the series zk BKeZk’.2, and its holomorphic periodic limit function g- satisfies the inequality Ig-(z)l 5 ylOi(y) (with yl = ~ K I U ~ O ~ ~ , ( & Y ~ ) ) , so that g E Q;. . (ii) For an application to the next lemma we formulate the second part, somewhat pedantically, as followo: Let A, , v E N, be finite subsets of -Yd such that (a) d(, c (b) U v A v = yd . Set h; = 2 pKeik“z, and h, = eze12h;. For any choice of

the sets -&I, , lim,-= h, = g in the topology of (5’. In fact, for every z, lim, h, (z ) = g(z), and Ih;(z)l s y16 t (y ) , so that lhvl-v y1 or lhvlo 9 y1 according as p & 0 or ,u < 0.

k e d V

LEMMA 3. (Uniqueness o f the Fourier coeficients.) Let (4”) be ajixed sequence of sets as just described, ,Bk ( k E Yd) a set o f coeficients, h; = 2 ,5keiR‘“, and h , =

k d l ,

h;ez212. If the sequence {h,} has a limit g in the topologV of (5‘, then g- is periodic, and

B K = [g-I,.

80 V. BARGMANN

Proof: Fix k. For some y o , k E Avo ; hence k E A, if v 2 y o . I t follows from the corollary to Lemma 2, Section 7.2, that [h;], = Pk if v 2 v,, . On any bounded set, g- is the uniform limit of h; ; hence g- is periodic, and [g-Ik = lim, [h;ll, = p k , q.e.d.

7.4. Formulation on Y'. Let v be a periodic distribution, and g = Av. By Lemma 1, Section 7.3, g(z) = ezz/2g-(z) = CPk(eg2/2eik"z), and by 3, Section 1.13,

('?k) 9 eZ2/2eik'.Z = Tn14ek2/2A(iL, k) = 2-n/271-nf4ek2/2A

where tk(q) = e ik 'p . 'Thus

Since / Ik = [g-Ik and p' = 2-"I2p, one finds from 117.4)

xk = p-l A ( z , - ik )g(z ) d n x , z = x + ;Y,y fixed . Jw;

In terms of distributions our results may be stated as follows.

PROPOSITION 7.1. Let u be a periodic distribution, so that TZblv = u for every vector Then u has the unique Fourier expansion u = 2 ak tk , k E Sd , 1 o f the lattice 9.

tk(q) = eik'q, converging to v in Y'. The coeficients are

(7.5) z = x + i y .

Here p is the uolume .f the base cell of 2, W,' is the base cell of 2' = 2-1/22', and y is jixed but arbitrary. If g = Av E nip, then Iuk[ 5 yleg(k), with some constant y1 . Conversely any set {uk} so chosen that Iukl 5 ylOE(k) ( f o r some y1 and some p) determines a unique periodic distribution u with Fourier expansion u = C uktk , and g = Au E . Since 9 t k = (277) nJ28k , v has the Fourier transfoarm Fv = (271) n/2CukBk .

7.5. The case of a regular distribution. If v(q) is summable over W,, equation (7.5) is easily transformed into the conventional formula for the Fourier coefficients. Fix k , and sety = 0. Defining B ( x , q) = A ( x , - ik )A(x , q ) , one finds

P% =I wo ,[J-f% q)v(q) d"q] dnx *

- Now B ( x , q) = ce-ik"xO;l(q - d 2 x) = ce-ik" xO;2(x -- q / d ? ) , where k' = d / 2 k and Since e2i?ik'.l' = 1 , we have B ( x , q - 27r1) = B(x + 2 ~ 1 ' , q) = T-n/2ek2/2.

HILBERT SPACE OF ANALYTIC FUNCTIONS--I1 81

if I E 9, 1' = 2-'i21 E 9'. Thus one obtains from (7.1), in view of the periodicity of v,

By (1.19), the inner integral equals e--ik'q; hence

uk = p-l~w~-ik~gv(q) dnq .

7.6. A special class of periodic distributions. Let vo E 9''. Then v = zle2 T2n1vo is a periodic distribution (generated by vo) if the series converges in Y'. This is always the case if v o has compact support. In fact, let g o =

Avo E E-p and g = Av. Then A-(T,,,v,) = g;(z - Z d ' ) , and g-(z) = z g ; ( z - 2 7 4 . The support of uo is contained in some ball (141 =( Y), so that, by

Proposition 6.1, 1

lg;(z)i =( re;(z)ec'E' yec2/2e;(z)eze/z 5 rle;(y)e[v, ( x ) ,

where c = d?, y1 = &lyecai2. Hence lg;(z)l ylO;(y)OiJ(x), and

- 2 4 (= rle;(r)ey;;(x - 2n17 .

As in the proof of Lemma 2, Section 7.3, we conclude from (7.3) that Z g;(z - 2971') converges absolutely and uniformly (on bounded sets) to the pqiodic function g-(z), that g E Q ; p , and that T2nlv0 converges in 9'' to v.

For the Fourier coefficients one finds

(ii) uk = p-l A ( z , -ik)g,(z) d"x = p-' A ( z , - i k ) {v , , xz} d n x . sin s; fl

In (i) and (ii), z = x + b; y is fixed but arbitrary. If v o is regular and vo(q) summable over R 71 , then

82 V. BARGMANN

EXAMPLE. Choose uo = 6. Set, in (ii), y = 0. Then { u o , x,} = 2 4 ( x , 0) , and uk = p-l. One obtains the well known formula z1 62r1 = p-l Zk eik.q .

8. Homogeneous Distributions

A distribution u is homogeneous of order il if it satisfies Euler's equation

where il may be any complex number ( [ 6 ] , Volume I, Chapter 111, 9 1.1). For its Fourier transform Fu this implies that l S v = 9 ( q . 8 ~ ) = -8 . q ( s l J ) (by (5.9)) or

(8.la) q - a(Fv) = - (n + A ) F u .

A well known example is the distribution

which is regular if 9 e il > -n and which may be defined by analytic continuation for other values of A. In addition, rA is orthogonal-invariant, i.e., P,rA = rA for every real orthogonal transformation 0.

In the first part of this section the distributions rA are studied by means of an analysis of their transforms f, = Arl. The second part deals with a classification of homogeneous distributions with respect to the orthogonal group. The main analytic tool is the use of harmonic polynomials and the corresponding decomposition of 3 and of its various subspaces. The distributions yL appear then as primitive building blocks.

8a. Orthogonal-Invariant Distributions

8.1. Formal preliminaries. (a) For any complex CI and any k E N we set ( E ) ~ = I?(% + k ) / r ( a ) , SO that

(b) In this section we write z2 = w. We recall that the Laplacian of an analytic function h(w) may be expressed as


For any positive integer k , A(wkj = 4k(k - 1 + $n)wk-l, or

(8.3a) 4 ( w X / a n , k ) = wk-' /Gn,k-l > cn,?; = 4'k! ( i n ) k .

Note that ( T , ~ , ~ = 1, and crl,k = ( 2 k ) ! . 8.2. Orthogonal-invariant functions in 3. Throughout this section 0

denotes an element of the real orthogonal group O(n, R). L e t f e 3, and let f = zkFk be its expansion in homogeneous polynomials of

order X- (i.e., pk E )uk ; see Section 3.6). Then V, f = zk V,p, . Hence V , f = f if and only if I.'& = pn for all k , and f is orthogonal-invariant (ix., V, f = f for all 0) if and only if 1 ) pk = 0 for odd k, 2) pk = c l p l if k = 21, c l L = const.

The relations (8.3a) suggest writing an orthogonal-invariant f in the form f = zk ykwk//a,,k so that

(8.3h) ' l f = 2 Y k t t w k l b n s k *

It follows that htf(0) = y l , and hence

(8.3c) f = z: @,.ld A m *

Rrmark. If n 2, it suffices to assume that V,f =ffor all 0 E SO(n, R) (the group oforthogonal transformations ofdeterminant 1 ) . Onlyif n = 1 is it necessary to require invariance under the full group (which consists of 1 and - l ) , and an orthogonal-invariant function is then merely an even function.

8.3. The distributions r'. (See [GI, Volume I , Chapter I, 4 9, and [ l l ] , Volume I , Chapter 11, 4 3.) For complex I,

defines a honiogeneous regular distribution which is orthogonal-invariant ({P, fov) = {r", y } ) . In addition it is analytic in A if i%x A > -n, i.e., for every

y E ,Lo, { r l , y} is analytic in I (see Section 4.2). I t suffices to observe that IrAy(q)l is uniformly bounded by the summable function ( (y((f(rY1 + r")@k(q) if --n < yl % i! y 2 , and k > n + y z .

THE MAP ArA. Let f, = AP. Then

i.e.,

(8.4a)

84 V. BARGMANN

(see (A.12)). Thusf,(O) may be extended to a meromorphic function with simple poles at 1, = -n - 2k, k EN. The poles are removed by setting

(8.5) un,a = (B,(1))- lra > g n , A = Avn,a , gn, , (O) = 1 >

and we shall study the analytic continuation of u,., or ofg,,, .

conditions are obtained for gn, , (see Sections 5.5 and 4.7) : As long as 9 8 1 > -n, v n , a satisfies Euler's equation (8.1), and the following

*

(8.6) (a) gn,a(O) = 1 > (b) V0gn.A = gn,a 9 (c) 4 " . agn.2 = 1gn.n

SOLUTION OF (8 .6 ) . We drop the restriction 9 4 1 > - n and show: for every complex A the equations (8.6) have a unique solution gn,A E 3. It follows that any orthogonal-invariant homogeneous distribution of order A must have the form uA-lgn,A, cc = const.

By equation (4.15),

9 " . 2 = 2 (?, & = 4 2 (d, + ZJ (d, - z,) . ..

Set g;,, = efw/2g *,,, and z * d = E zj dj . Then by (8.6),

(8.7)

As indicated in Section 8.2, we write g:,, = zk y ; z f k ~ k l ~ n , k . 2kwk we obtain, using (8.3b),

2 d,(d, - 2z,)g:,, = 2Ag:,, or Ag:,, = 2(1 + n + z . d ) g l , , . Since z * dwk =

rI,k+l = 4(k + + ' ) ) y t , k > yT,k = + n))k 9

(8.8a) g:,A = lFl(=k(A + n), in, W ) 3 gn,, = e-"/'lF1(fr(1 + n), 4n, w) *

Here ,Fl is the confluent hypergeometric function

Similarly one finds Ag;,A = 2(1 - z . Y ; , ~ =

(8.8b) g;,a = ,F,(-$A, in, - w ) , gnBn = ew/21Fl(-&~, an, -w) . Comparison of (8.8a) and (8.8b) shows that gn,n( -w) = g,,-,-,(w). This

implies that, for every complex 1, gn,, E E'.

Proof: By construction, gn,A E 6' if 9 4 1 > -n. Since the transformation w -+ -w (equivalently, z -+ -iz) preserves the norms I . I , , it follows that gn,, also belongs to 6' if 9,( -1 - n) > --n, Le., 9 4 A < 0.

Consequently we may extend the definition of vaz.z to all complex A by setting v n . 2 = A-lgn.2 *


FOURIER TRANSFORM. By equation (5.7), ($g,,,)(zu) = g n , J - w ) . Thus we find for all I,, in accordance with (&la),

(8.9) $ g n , i = gn.-a-n 7 g u n , , = u n , - i - n .

ANALYTICITY OF u ~ , ~ , (OR g,,J. For every f E @, (f, g n , A ) is an entire analytic

Proof: For 9 8 1 > -n the analyticity has been established (in the If W E 1 < 0, then (5 g n , a ) =

function in 1 (i.e., u n , , is analytic in the whole A-plane).

first paragraph of the present subsection).

q.e.d. (J$gn, -a-n , \ - - , /S-l f, gn,-n-n) is holomorphic because We ( -3 . . - n) > -n,

8.4. Further discussion. If 9 k 2 > 2 - n, then by (8.5)

(8.10a) = Jun,a--2 , (8.10b) q 2 u n , A = (1 + n)un.l+2 >

where A, = zj a2/aq3 , the symbol h being reserved for the Laplacian zj a2/azg. By analyticity these equations extend to all 1. (They also follow from (8.8) and (8.3b), since they correspond to Ag;,, = 2Ag;,n-2 and As:,, = 2 ( A + n)g: , j+2 or Y ; , , ~ ~ = 2Ay;-,,, and yT,lc+l = 2(A + n)yT+,,, , respectively.) In addition,

(8.10~) A, log = Bn(OfUn,-2 9 /?,(0) = (47r)n'a.

In fact, log r = a,rA l a=o , with 2, = 2/81. Hence,

log = a a ( B n ( 1 ) A p u n , a ) l l = o = ~ ~ ( A B n ( J - ) u n . ~ - J (,?=a = Bn(O)un,-z *

\tre conclude from (8.10) that every v ~ , ~ may be expressed by regular distributions as follows. 1) If &?e 3, > -n, then u ~ , ~ = (/?n(l))-lrA, and ra is regular. 2) If 92% 3, -n,

where s is a positive integer. If n a suitable s such that -n < 2s + ,!i% 1 < 0. valid unless 3. is an even negative integer.

3, one may always use the first expression for If n 2 2, the first expression is

DETERMINATION OF u , , ~ ~ , ?.k = -n - 2k. 1 ) By (8.8a), gn,-n = e--pc12 = ,n%,

so that ZJ,,-~ = ~ r ~ / ~ 6. It follows from (8.10a) that

(8.1 1) vzz ,Ak = (-2)'k! 0;,9r"/~ At 6 , Ak = -n - 2 k .

86 V. BARGMANN

2) The residue of rA = ,Bn(A)vn,, at A, is therefore

Res ra I lk = cr;;,\wn A: 6 , An = -n - 2 k ,

since, at A,, I ' ( g ( i + n ) ) has the residue 2 - ( - l ) k / k ! . (Thus {TI, y> is a meromorphic function of A with-at most-simple poles at Ak and residues -

LDn A:Y(o ) /bn ,k ') From (8.10) and (8.5) one obtains the classical relations

A,+" = wn7rnJ4 A,vn,,-, = (2 - n ) w , S ,

and A, log r = w,S if n = 2.

BOUNDS FOR g,,, . I t follows from Theorem 4.2, that, on every bounded set in the A-plane, g n a A is bounded in some norm. More specifically, if -n - p 5 228 i p and 19~ A) 5 p, then Ign,l)-p 5 bn,p for a suitable constant or, equivalently,

8.5. Remarks on , F , . An alternative method of deriving (8.8a) is the following. Insert in (8.7) the formula (8.3) for the Laplacian and 2wd/dw for z d. Then R(A+n),2,n,2gi,, = 0, where

(8.13)

If b + k # 0 for all k E N , then the equations f (0) = c, Ra,bf = 0, have the unique holomorphic solutionf(w) = c lFl(a, b, w).

Note the identity eWIFl(a, 6 , -w) = ,F,(b - a, b, w), for which (8.8a), (8.8b) provide an example.

(Proof For the derivatives one obtains

For g(w) = eMIFl(a, 6, -w) one finds g(0j = 1, R,-,,,g = 0.)

(8.13a) d k - F (a, by w) = gk1Fl(a + k, b + k , w) . dwkl ' ( b ) k

POLYNOMIALS. ,F1(a, 6, w) is a polynomial of order k if and only if a = -k. In terms of Laguerre polynomials

(8.13b) ( y + 1 ) k iFi(--k, y + 1, W ) = k ! Lg(w) .

** If W e A )= 0, one may use the estimate (1.1 Id) and a similar estimate applies to the interval 0 2 9 8 2 2 -&z. Since the Fourier transform preserves norms, it follows from equation (8.9) that lgn,II-p 6 n , p as asserted.


The expressions for g n , A yield some well known relations for Hermite polynomials. By (8.8a), g;,-n = cW, g&+ = e-" ,F,(-k, &z, w) , and by (8.10a), 4 k -

(8.14) eU'(Akcza) = (-4)k(&n)k1F1(-k, in, w ) = (-4)'k! Lp-z)'z(w) . If n = 1, then elu(akP) = H,k(z) . Since dH,,+,/dz = 4(k f l)Hzk+, (see (A.22)) , one obtains from (8.13a)

(8.14a)

(See, for example, Szego [ 111, Chapter V.)

g n , - n = ( - 4 ) k ( i n ) k g n , - n - 2 k . Thus

H2k+l(z) = ( -4)k(~n) ,2z ,F1(- -k , 8, w) = (-4)'k! ~ Z J $ ~ ( W ) .

8b. Classitication of Homogeneous Distributionsz5

8.6. Harmonic polynomials and the spaces 3,. Throughout this subsection we assume n 2.

1) Harmonic polynomials of order &denoted by Y,(z)-are homogeneous polynomials of I-th degree which satisfy the Laplace equation A Y , = 0. They form a linear subspace 'p: of 'p, (see Section 3.6).

It is a basic fact that every polynomialp E !& has a unique decomposition:

(8.15)

(For a simple proof, see van der Waerden [13], p. 13.) This implies for vi = dim 9;' (8.15a) v b = 1 , v i = n , V ~ = V ~ - V ~ - ~ if 152, v,=dirn!Jlp,.

For any positive integer k and any Y , E @ ,

A(wkYY,) = 4k(k - 1 + &n + I)wL-'Y,,

as is readily verified. In analogy to (8.3a) we may write

(8.16) A(wky t /on+21 ,k ) = wk-lyl /Gn+Zt,k- l '

2 ) If 1 # 1', then ( Y z , Yi.) = 0 because Y , and Yip are homogeneous polynomials of different degrees. Consider next u = (W'Y, , wk'Y;,), and assume (X f 0, which implies that 2k + I = 2k' + I ' . Let k' 2 k. Since zj and d, are adjoint (see end of Section 4.5), u = ( Y , , Ak((wk'Yi,)). By (8.16), Ak(wk'Yl'.) = 0 if k > k'. Thus k = k' , 1 = l', and 0: = C I , + ~ , , ~ { Y ~ , Y,').

25 In this part a number of derivations are omitted or only briefly sketched. Most of them, however, are fairly straightforward.

88 V. BARGMANN

Choose in every !# a fixed basis of orthonormal elements Y , , , 1 2 t v; , i.e., such that ( Y l t , Y L f , ) = d,,. . l h e n we have

It follows that the elements wkY,, are pairwise orthogonal in every 5* and that Ilw”Y,tIIi = ‘,+2l,k$+2k (see (3.13a))’

3) A dzferentiation formula.26 For any holomorphic g(w)

( Y , ( d ) is constructed by substituting the differential operators d j for the variables ti.) Equation (8.18) is a special case of Hobson’s differentiation formula, but it is easy to obtain it directly by induction on I , the case I = 0 being trivial. (Let I > 0, and assume (8.18) for 1 - 1 . Set X , = diY, so that LY, = I 9 z j X j , and l Y , ( d ) g = zi d,(X,(d)g) . Since X j E YE,, X , ( d ) g may be computed from (8.18), and the desired result follows.)

4) For harmonic polynomials Y,(q) of real variables-equivalently for “spherical harmonics” r-’Y,(q)-an inner product is naturally defined by an average over the unit sphere in R, ,

(8.19)

(do(q) = Euclidean measure). One finds that

Proof: For yo = ~ - ~ f ~ e - ~ ~ l ~ , Avo = 1 and Afp,, = eu12, by 5, Section 1.13. Hence,

A+( Y,(q)yo) = 2-z’2Y,(d)eW’2 = 2-r/zYL(z)ew/2 ,

by (8.18), i.e., A( Y,p,) = 2-’12Yl(z). Thus, with s = g ( l + l ’ ) ,

and (8.19a) follows (from on = 2 ~ ” ’ ~ / r ( i n ) )

p 6 See Hobson ”31, p. 124.

HILBERT SPACE OF ANALYTIC FUNCTIONS-I1 a9

\Ve mention that A-l maps the functions w k Y z t / Z / d n 1 2 1 , k into the complete orthonormal system

on !?J,~ = L2( !Rn) (see (8.13b)). Here y = 1 + +n - 1. cij The spaces 3’. Let f = C p , E 3, p s E 12,. Every p , may be decomposed

according to (8.15) so that, in terms of the basic functions Y,, , p s = C ak,twk1711t, I + 2X; = s, with uniquely determined coefficients M . Hence

As can be shown, the series converges absolutely and uniformly on bounded sets in

C m

. Finally, setting f i t = 2 akltwk, we obtain k=O

To express the coefficients x by the derivatives off a t 0, note that, by (8.17),

bn4ZZ,kakl t = (wky,t > P s ) = (1, y ~ t ( d ) AkPs> = P,t(d) AkPs(0) *

Here s = 1 + 2k, and P,, = KY,, (Section 5.1). Since, at z = 0, the s-th order derivatives off and of p , coincide,

(8.20b) ClkZt = (‘n4-2L.k)-’PZt(d) A”fo) *

Let sz be the linear subspace of 3 containing all functions of the form g =

h , ( w ) Y, , , the functions h, being necessarily holomorphic. Equation (8.20aj

is then the decomposition of any f E 3 into its components in 3L (3’ contains all orthogonal-invariant elements of 3).

Every 3p admits the corresponding decomposition 5” = C @ ~ ” ~ ‘ into the pairwise orthogonal subspaces C,p ,L = 3p n 3’.

\Ye mention the following. I f f € 5p (@), then in (8.20a) f i t E 5p+z (WL) for every 1 and t .

6) Group-theoretical remarks. The decomposition into the subspaces 3z has primarily a group theoretical significance. Every CpF is invariant under all operations V , , and the operators Yo restricted to q; define an irreducible representation, say 9;, of O(n, R). Moreover, two different representations 9; and

v;

t = l

90 V. BARGMANN

9:, 1 # I f , are inequi~alent.~' (For n 2 3 this remains true if O ( n , R) is replaced by SO(n, R ) . For n = 2, different 9; remain inequivalent, but they are no longer irreducible if restricted to the Abelian group SO(2, R) , excepting the trivial case

Similarly, ~ ~ ' $ 3 : ~ i.e., the space of all w k Y , , is invariant under all Vo and carries the representation 53;. (Thus, for example, i r i P r 1 is decomposed into the pairwise orthogonal subspaces each of which carries the same irreducible representation of O(n, R). )

As a simple consequence of Schur's lemma we have the following proposition : Let T be a linear operator mapping 3-or any of its subspaces which are invariant under Vo (such as Q, Q', BP)-into 3, and assume that TV, = V,T for all 0. Then f E sz implies that Tf E 3,. More precisely, if, for fixed g (E 3O) and 1, gY,, are in the domain of T, then T(gY,,) = hY,, (for all t ) where h (E 3') depends only on I and on g. Note that no further restrictions are placed on T.

In essence our discussion of homogeneous distributions is based on this proposition since the operator T = * 8 by which these distributions are defined satisfies the hypothesis TV, = VoT. The proposition, however, will not be used explicitly; the relevant results will be obtained by direct construction.

7 ) The operators M , . For any f E 3 we define, in terms of the decomposition "1

(8.20a), M , f = 2 f l t Y 1 , . Evidently the linear projections M , satisfy the

following conditions: (a) M , 3 = sz, (b) M , f =J' i f f E 3', (c) M l f = 0 if f E 3" and 1' # 1.

1 = 0.)

t=1

Using (8.20b) one obtains M , f = z:k,t u,,,wkYy,, , i.e.,

m for all f E 3. If 1 = 0, Ado f = 2 c;,lpk A"f(O), and equation (8.3~) appears as a special case for f E 3'. k=O

Representation theory yields a second expression for M , :

(8.21a)

Here dO denotes invariant (Haar) integration on O ( n , R), the integral extends over the whole group (whose measure is normalized to unity), and xf is the character of the representation 9; . (For 1 = 0, v; := 1 and x: = 1.)

2 7 This may be established in a fairly simple way by induction on the dimension n. One may of course also use the results of representation theory. Harmonic polynomials of order 1 can be identified with symmetric tensors (of rank I ) with vanishing traces. In Weyl's notation the representation 9:, n 2 3, is characterized by the following choice off,: fl = I , fa = . . . = 0. (See [14], Chapter VII, 0 9.)


It is easily shown that every M , is a properly bounded operator on (5’ and that M: = hf, (see Sections 4.4, and 4.5). On every sp, M , is the orthogonal pro- jection onto 3 P , , .

In the notation of Section 3.13,

M, = vi fxt(0-l) V, dO , Mo =/vo d o .

Thus Mafis simply the group average of V0$

8.7. The case n = 1. If n = 1, the decomposition into the subspaces 3, degenerates into the trivial decomposition 3 = 3O 0 3l, where 3’ and 3’ consist of even and odd functions, respectively. The formal machinery, however, remains applicable, and it is not necessary to treat the one-dimensional case separately.

In fact, here A Y , = d2Y,/dz2 = 0 ; hence 1 = 0 or 1, vi = v; = 1, and there are only two functions Y, , , namely, Yo, = 1, Y,, = z. Equation (8.21a) reduces

1 =0 , 1 *

to ( M , f)(4 = t ( . f ( z ) + ( - l ) Y ( - 4 ) 5

8.8. Homogeneous distributions. Let q * a v = Iv. Set f = Av, and f+ = eu’’2f. Then ( g * 2 - A)f= 0 or, by equation (8.7),

R , f + = ( A - 2 ( 1 + n + ~ . d ) ) j + = 0 .

and

so that, by (8.13), B A ( ~ Y , ) = 4 ( R ( ~ + , + , ) , ~ , ~ + ~ / z h ) Y ~ .

Writing f = C f l t ( w ) Y , , andf+ = Z f $ Y , , (f: = ew’2 f i t ) , we have therefore

B,f+ = P A ( f t , Y , t ) = 4 z: (R(nsr+,),z.l+n,2f$)Ylt l , t l , t

Thus B , f+ = 0 ifand only if Rcl+n+,)/2,1+n,2f$ = 0 for all 1 and t-equivalently if and only if

f t t (w) = Elf 1&(B(n + n + 4 1 + 4% 4 = w : + 2 , , , - 1 (4

(see equation (8.8a)) with some constant m l t . Setting Et M , ~ Y & , = Y , we may state the following result:

92 V. BARGMANN

A distribution u is homogeneous of order 31 zfand onb ;ff = Av has the form

(8.22) f ( Z ) = Z I & ( w ) Y , ( 4 3 gF,,,(w) = gn-72z.1-1(4 .

Since Y 1 ( - i z ) = i - 'YZ( z ) , and g L , A ( - ~ ) = gk,-n.-,(w), by (8.9), one obtains for the Fourier transform

(8.22a) ($3 ( z ) = 2 i-zg:,-n-,(4 Y , ( Z ) ' 1

ANALYTICITY OF gi,AYz . It follows from the estimate (8.12) that, for fixed Y , , g f , ,AY , defines a family of distributions which, in the whole d-plane, is analytic in L.

Remark on the one-dimensional case. For n = 1, equation (8.22) solves the problem mentioned in 4, Section 5.6. For every A the equation ( q . a - 31)g = 0 has precisely two linearly independent solutions in E', namely, g,,,(z2) and Z ~ ~ , , - ~ ( Z ~ ) .

8.9. Connection with the distributions r'. I t remains to relate the distributions just obtained to 7'. Let Y , be a harmonic polynomial to be keptjxed. Then the following two distributions are homogeneous of order 2:

(8.23) (a) Y1(q)rk-- l , (b) Y,(a)rA++l.

-

At first we assume 94~ I > 1. The differentiation forrnula (8.18) is applicable to (b) and yields

by (8.18), (8.13a) and (8.22). Inserting the value of /I,, we obtain, for W e J > 1,

Set for all d


so that u ; , ~ is a holomorphic family of homogeneous distributions. Then, by (8.24),

(8.2 5a) Y1(q)rL-z = P n * L ( 1 ) 4 , A if 9?e 3, > 1. This equation also defines a meromorphic extension of Y,r"-l with simple poles at A = 1, - 1 = -n - 2k - I , k E N (see Section 8.4).

\Ve note the following relations:

(8.26a) A,uf,, = (1 - OUf,&2 Y

(8.26b)

They follow from (8.25a), (8.23c), and (8.5) for 928 1 > I , and by analyticity for all 1. For the Fourier transform one finds SV; ,~ = i-zuk,-n-L (see (8.22a)).

8.10. Further discussion. 1) Determination O ~ V ~ , , , ~ . If 1 = 1, - I, then (i(1. - 1 + 2)), = (-1)"(&(n + 2 k ) ) , . Hence, from (8.11) and (8.26b),

2 = ( o n + Z l , k ) - 1 ( - 1 ) ,+12,-1/2k! r"/4yl(') A: 6 Un, h- 1

and one obtains the residue

(8.27)

- 2) It is instructive to examine the integral I(1, y ) = jY,(q)ia--ly(q) dnq

directly ( y E 9). While 1(1, y ) , in general, fails to converge absolutely if 928 1 -n, its principal value I,,(& y) is defined for W e 1 > -n - I , and Io(A, y ) also furnishes the analytic continuation of the distribution YLrA-z, including the residues (8.27).

Set u i ( q ) = rA-lY,(q) and, for 0 < e < 1,

u1, w) = j- d"q > 441, Y ) = lim U J , w) E - 0

Id P i \\'rite

where 1AA w) = y ) + v, y ) >

~ ~ ( 1 , y) = uA+ dnq and b(1 ,y ) = uA+dnq. E S I P I 21 t a t 2 1

The integral b ( 1 , y ) is absolutely convergent for every 1, and b(1, y) is holomorphic in A.

s As usual we separate from y a section of its Taylor series:

y = 5, + ys, C s ( 4 ) = 2 u [ " ' ~ [ m l Y ( o ) l [ m ! l Y

Iml 5 s

94 V. BARGMANN

V'

Assume Y , = 2 PtY,, . Then t=1

If s 1 - 1, the inner integral vanishes (as follows from (8.15) and (8.19a)) so that a e ( l , 5,) = 0, and Io(l, y ) = u o ( l , ys) + b(1, y ) is holomorphic if 9?e 1 > -n - 1 as asserted.

In order to evaluate ~ ~ ( 1 , 5,) we set 5, = G L X ~ ~ , ~ J ~ ~ Y ~ ~ , , 1' + 2k s. According to (8.17) and (8.19a),

so that

If W e 1 > -1 - n, then ue(A, 5,) converges to a,(& c,), and

(To obtain pk apply (8.20b) to 5, and use the fact that a c m l ~ , ( 0 ) = &"]y(O) if Iml s 5 . )

SUMMARY. (a) If B?e 1 > - n , then { Y L y A - ' , y> = I(A, y ) . Here I(1, y ) is an absolutely converging integral which is holomorphic in A, and I ( 1, y ) = I,( 1, y ) . (b) Io(L, y ) remains well defined and holomorphic as long as 9 e L > -n - I, and it admits then for every s the decomposition

(c) The right-hand side of this equation defines a meromorphic analytic continuation of I , into the half-plane W e 1 > --n - s - 1. Only the first term u o ( l , 5,) has poles (at 3Lk - I, see (8.28a)), and the residues pk are consistent with (8.27). (d) For the distributions v : , ~ we find: 1) If :%e 1 > -n - I, { u : , ~ , y } = (/ln,z(il))-lZo(A, y ) . 2) If .C~?E A < --I, we may use (8.2613) because then g: # 0. --

HILBERT SPACE OF ANALYTIC FUNCTIONS-I1 95 -

Thus {u:,~, y } = (-1)'(a:)-l{~~,~+~, P,(a)y}, and integral representations for

u n,

8.11. The one-dimensional case (see Section 8.7) is of course well known. Here we deal with the distributions lqlA, 1 = 0, and q . lqlA-l = (sgn q ) lqlA, 1 = 1. Equation ( 8 . 2 3 ~ ) reduces to

I have been determined in Section 8.4.

d IqlA++l/4 = (1 + l)(sgn 9) 1qIA .

To transcribe (8.27) we note that w1 = 2, = (2k) !, and = (2k + 1) !. Thus, with 1, = -1 - 2k,

~ 3 ~ ' " ) denoting the m-th derivative of 6.

Appendix

AI. Scheme for a convergence proof. The following elementary scheme is used on

Given a sequence {ai} and two double sequences {bij} and { c i j } of non-negative real several occasions and is formulated here for easier reference.

numbers such that

(i) bij < E for all i i f j 2 j,,(E) (uniform convergence), (iij cij < E if i 2 io( j , E ) .

If ai 2 b i j + cij (for all i , j ) , then limi ai = 0.

1. + 8. = 8, q.e.d.

continuous variables. 2. The scheme clearly applies if ai 5 max (bi j , c i j ) .

Proof: Set j , = j O ( Q e ) , and i, = io(jl, 4 ~ ) . Then, for i 2 i, , ai 5 b i , j , + ci,jl <

1. There is an obvious generalization to the case where i, j or both are Remarks.

AIL The functions 5 (see Section 1.4).

a. BASIC INEQUALITIES. 1. For u, v E cn, (A.1) 1 + IU + .I2 5 K 2 ( 1 + lUI2)(1 + l V l 2 ) > K = ($ ) ' I 2 .

Proof The difference

A =i 4(1 + lulz)(l + 1 ~ 1 ' ) - 3(1 + lil + 4')

may be written as the sum of three non-negative terms,

A = I U - uI2 + 12U. u - 1 l a + 4(luf2 1 ~ 1 ~ - (U . ~ 1 ' )

Equality holds if u = u, /uI2 = 4.

96 V. BARGMANN

2. For any two vectors a, b E c, and 0 < 1 < 1,

I 1 - 1 (.4.2a) la + b12 2 I la12 - - lbI2 Y

(A.2b) la + bI2 I- $ la\’ - Ib l2 , 2 ’ I = 1.

3 . If the vectors a, b, c satisfy the equation a + b + c = 0, then by (A.l)

1 + la12 s 271 + lb12)(1 + Icl2) , 1 + lb12 5 ,a(l + Ia12)(1 + lc12)

Hence

(A.6b)

b. FURTHER INEQUALITIES. 1. For a complex number y # 0 set ,!I1 = min (/?I, 1yI-l) and p2 = max ( Iy l , 1rl-l). For real p one readily finds

(A.7) pple; (a) s qya) 5 p ! $ ~ ; ( ~ ) . 2. If u > 0, then 0Fa(s), s E R,, is bounded. Its maximum T; is

T; = O;.(O) = 1 if p 6 u ,

(‘4.8)


Proof:

c. IUTEGR~L INEQUALITIES. 1. Dq5nition. Let

@ , ( u i 1 9 , ~ ( u ) = OFf.,(ui 5 T Z - ~ .

(.4.10) s = T-"" 6;'(q) dnq , a g o .

(The integral converges (a) if a > 0, (b) if GL = 0, p < -n.) For any a E Rn and 0~ 2 0,

Proof: BY (x.3), e;(q + a ) 0 T U ( q ) s . l ~ l e ~ ( ~ ) e p , , ( q ) e d ( ~ ~ = K I P I o ; ( a ) e ; ; I P I ( q ) .

2. The case a > 0.

(X.lla) -it,- = (Z/a)"",

(h . l lb1 j f , n < ~ f ~ ( 4 / a ) ~ / ~ .

Proof: (a, ;;," is a Laplacian integral. (b) Since O;'(q) = e;a'2(q)0,a'2(q) 5

3. Set v = q2 and introduce polar coordinates. The Euclidean area of the unit sphere

Ta/Z,+2 ( q j , we havejf,, < ~;'"j!i .

(141 = 1; equals

(.4.12) w, = 2 x n / 2 / r ( ; 4 .

Hence, if (x 0,

(A. 12a) jz,fl = (r(&nj)-l e-a"'2(1 + v)pi2v(n-2)/2dv. r (Note that this holds also for n = 1, with the choice w = 2 in accordance with (A.12j.)

4. The case GL > 0, p > 0, n 2 2. Here

05

e-aui2U(~+n-2)12 d u < r(gn)GSn gS, e-a8/2(l + ,)(p+n-z)/zdu. 1% Setting UJ = 1 + D , one obtains for the right-hand integral

e a : 2 1 e e - a " ' Z 1 1 1 ( p + " ~ z ) i z dw < ea/2 e-aw/2W(p+n-z) /2 dLe, . Ix 'Thus

(A.131 (2/ai(p+n)izr(q(p + n)) < r(&n)j;,n < ea/2(2/~)(P+n)i2T(~(p + n ) ) . 5 . The case a = 1, p > 0, n = 1. For ti = 1 we mention, without detailed derivation,

the rough estimate

(A. 13a) 2(P+1)'2r(B(p + I ) ) < r(B)& < e1122(p+*)'2r(4(p + 1))

which is used in Section 1.12.

98 V. BARGMANN

6. The case cz = 0, p < -n. We need here the formula

(A.14) m

IN,v =I u"-l(l + u)-v dv = B ( p , 'V - p) , v > p > O ,

where B denotes Euler's Beta function. (Setting = v / ( l + u ) we obtain IN," =

p ( 1 - QV-f l -1 dt.)

Let p = -0, 0 > n. Then

(A.15) . L , n = r(t(a - n))/r(+) . Proof: Let /3 > 0. Then

By (A.12a) and (A.14), I'(gn)j?,,, = In12,012 := B(&, - n ) ) .

(A.16) F " / ~ J ( / ? ~ + q2)-"'2 dnq = /?n-"jp_,,n .

7. Integrals over E n . These correspond to integrals over R,, . Thus

(A.17) n

Since Qn(z) = nn e-lz12 dnz, one finds for example

AIII. Hermite polynomials.

a. DEFINITION AND GENERATING FUNCTION. 1. For any multi-index m define :

(A.18) e - Z 2 f f m ( z ) = ( - 1 ) l m l dim] e--2', Z E C , .

If rn = (inl , * * * , m,) and z = (zl, - * - , z,), then

(A. 18a)

2. Generating function G ( t , z ) . Let t E Ea . The Taylor expansion of e-(z- t )z (in t ) is e-za zm t [ " ] f f m ( z ) / [ r n ! ] . Thus

(A.19)

Expand both factors of G, e-ta and e 2 t ' z , in powers of t :

(A.20)


where (2z)Ikl = equations j

( Z Z ? ) ~ ? = 21kl~[k1. Similarly, the relation 2 t . z = e@G(t, z ) yields the

(A.21:

expressing the monomials zrm] as linear combinations of Hermite polynomials.

b. THE MODIFIED HERMITE POLYNOMIALS H , . DeJnition:

ezz&m(z) = dcm] eza .

It is readily verified that k m ( z ) = ilmlHm( - iz) . In analogy to (L4.19)-(A.21) we have

(A. 19a)

(A.20a)

( A 2 1 a)

c. RECURSION RELATIONS FOR n = 1. From the identities (i) aG/az = 2tG, (ii) .3G/at = 2 ( z - t ) G one obtains the classical relations

(i2.22)

(.A. 2 3 ;I Hl = ZzH,, Hm+l = 2(zHm - mH,-,) f m h l .

d. ESTIMATES FOR n = 1. The following inequalities hold:

dH,.+Jdz = 2 ( m + 1)H, , dH,/dz = 0 ,

(.4.24) I H ~ ~ ( Z ) I 2 2 2 , ~ e;,(z) , I H ~ ~ + ~ ( Z ) I 5 22s++'(s + I ) ! it1 e;s(z) . They are obvious for s = 0 (H, = 1, Hl = 22) and follow for all other s by induction from (A.23) .

Replace the second inequality in (A.24) by

IH2s+l(z)I 5 22s+1(s + I ) ! Ols+l(z) . Then for all m

(A.24a) IH,(z)l 5 Zm[+(m + I ) ] ! eL(z j , Z € C ,

where [ [ I denotes the integral part of the real number E , i.e., the largest integer not exceeding 5.

e. ESTIMATES FOR ARBITRARY n. From (A.18a) and (A.24a) one obtains, for all z E Cn,

(A.25) I H m ( 4 s 2'm'B,m,eo,,(Z) Y !k = [h(k + n)l! 9

by means of the inequalities: (iii) r]: kj! 2 (29 k ? ) ! , k j E N.

(i) zj [tj] 2 [zj t j] , (ii) O:(zj) 5 O:(t) if /A 2 0,

j

100 V. BARGMANN

List of Symbols and Definitions 28

1. SETS AND FUNCTION SPACES.

IR, R+, R,, C, C,, N (Section l . l ) , @+ (Section 1,2), Ck, C“, 9; (Section 1.5), Y”, 9 (Section 1.6), Y’ (Section 2.5), 3 (Section 1.7), Q; (Section 1.14), Ep, E (Section 1.15), E’ (Section 2.12), (Section 1.17), 9, 113 (Section 2.1), ps (Section 3.6), Bp (Section 3.8), 3 (Section O . l ) , 5” (Section 3,1), (sp)’ (Section 3.12).

2. MEASURES.

d”q ((1.7), Section 1.8), d”z, dp, ((1.22), Section 1.18), dpUP, ((3q1), Section3.1).

3. NORMS.

Jnl (Section l.l), 11.11; (Section 1.6), 1.1 (Section 1.15), J j - J J p (Section 3.1), Il.llb (Section 3.12).

4. INNER PRODUCTS ; HERMITIAN (SESQUILINEAR) FORMS.

a . b (Section l . l ) , ( , ) (Section 1.18), { , } (Section 2.5), ( , ) p (Section 3.1).

5. OPERATIONS.

drnl (Section 1.3), A, A+, A- (Sections 1.8 and 2.12), Q, (Section 1.17), W (Section 1.19), J b , JF (Section 4.5), V,, V , , V,, A, (Section 4.6), q, $ (Section 4.7), P = A-lVA, Ta,b, T, (Section 5.4), ‘Po (Section 5.7), 9, 5 (Section 5.8).

6. FUNCTIONS AND DISTRIBUTIONS.

qCml, zCml (Section 1.3), u, (Section 1.7), A, A+, it-, v m (Section 1.8), x,, e, , t , (Section 1.13), H,, H , (AIII), 0; (Section 1.4 and AII), e,P, tp (Section 3.8), 6 , 6 , , 8, 8, (Section 2.10), u k (Section 3.6).

A

7. CONSTANTS.

K (A.l), 3.7), a,.” ((8.3a), Section 8.1).

(A.8), CO, (A.12),& (A.10), v’sp (Section 3.6),cp , c i , c; (Section

8. OTHER SYMBOLS.

[mi, [m!] (Section 1.3),f+, f- (Section 1.7), M ( r , f ) (Section 1.15), sp(m) (Section 2.1), 0 (Section 5.10, l l ) , - (Section 1.16, footnote 6).

9. DEFINITIONS.

Multi-index (Section 1.3). Convergence in eP (Section 2.2), in (5 (Section 2.3), in a’ (Section 4.1). Fundamental set (Section 2.1), I-invariant set (Section

** In general only those symbols and definitions are mentioned which are used in several sections.


2.1), ,?-regular element (Section 2.1), analytic functionals (Section 4.2), f'unctionals on (5' (Section 4.3). Linear operators on (5 and (5' (Section 4.4), properly bounded operators (Section 4.4), adjoint operators (Section 4.5).

Bibliography

[ l ] .\ronszajn, N., Theocy of reproducing kernels, Trans. Amer. hlath. SOC., \'oL 68, 1950, pp. 337-

[2] Bargmann, I'., On a Hilbert space of analytic functions and an associated integral transform. Part I ,

[3] Bargmann, \'., On the representations o f the rotation group, Rev. of Mod. Phys., Vol. 34, 1962, pp.

[4] Bergman, S., T h e Kernel Function and Conformal Mapping, Mathematical Surveys No. V, .American Mathematical Society, Sew York 1950.

[5] Dunford, N. and Schwartz, J. T., Linear Operators, Part I . Interscience Publishers, New York, 1958.

[6] Gel'fand, I. M. and Shilov, G. E., Generalized Functions, Vol. I, 11, Fizmatgiz, Moscow, 1959, 1958. (In Russian.)

[7] Gel'fand, I. M. and Vilenkin, N. Y., GeneralizedFunctions, 1'01. IV, Fizmatzig, Moscow, 1961. (See also the English translation of Vols. I and IV: Generalized Functions. Academic Press, New York, 1964, and the French translation of Vol. 11, Les Distributions, Dunod, Paris, 1962.)

[8] Grossmann, A., Hilbert spaces o f type S , Journ. of Math. Phys. Vol. 6, 1965, pp. 54-67. [9] Hobson, E. LV., T h e Theor). of Spherical and Ellipsoidal Harmonics, Cambridge University Press,

1931. [ 101 Plancherel, hl. and Pblya, G., Fonctions entiires et inte'grales de Fourier multiples, Comment. Math.

Helv. Vol. 9, 1936, pp. 224-2.18; Vol. 10, 1937, pp. 110-163. [ 111 Schwartz, L., The'orie des Distributions, Hermann, Paris, 1957, 1959. [ 121 Szego, G., Orthogonal Pobnomials, Colloquium Publications, '4merican Mathematical Society,

[ 131 Van der LVaerden, B. L., Die Gruppentheoretische A4ethode i n der Quantenmechanik, Springer,

[14] IVeyI, H., The Classical Groups, Princeton University Press, 1946.

Received June, 1966.

404.

Clomm. Pure Appl. Math., 1'01. 14, 1961, pp. 187-214.

829-845.

New York, 1959.

Berlin, 1932.

bargmann, on a hilbert space of analytic functions and an associated integral transform

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