r. t. seeley - spherical harmonics

Spherical HarmonicsAuthor(s): R. T. SeeleySource: The American Mathematical Monthly, Vol. 73, No. 4, Part 2: Papers in Analysis (Apr.,1966), pp. 115-121Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2313760 .

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SPHERICAL HARMONICS

R. T. SEELEY, Brandeis University

The object of the present article is to give a concise and elementary exposi- tion of spherical harmonics, including the Funk-Hecke-theorem, and some esti- mates of their derivatives in absolute value and in L2 norm. By elementary, we mean independent of any knowledge of special functions. In fact, the first two theorems require only Green's formula for the Laplacian and a little linear alge- bra. Our proof of the Funk-Hecke theorem relies on approximation of an integrable function by a continuous one, and of a continuous function by a polynomial. Theorem 4, estimating the derivatives of spherical harmonics, uses Leibniz's formula for the derivative of a product in several variables, and Theorem 5, concerning the expansion of a C? function in spherical harmonics uses a density theorem. Thus this article may serve those who would like to use spherical harmonics without presupposing special functions, and to those who wish to present to their students some applications of the results mentioned above.

Theorems 1, 2, and 3 are covered in the standard reference [I], and Theorem 4b in [2 .

We consider the harmonics on Sn-1, the unit sphere in Rn, whose volume is denoted ( Sn- 1 x, y) is the inner product in Rn, and x 2= (x, x). We denote by A the Laplacian in Rn, Ag = 1 a2g/94

We show that the spherical harmonics are the eigenfunctions of a self -adjoint operator, the Laplacian on the unit sphere. In general, a smooth hypersurface S in Rn has a Laplacian which is derived from A in the following way. Given any function f on S, consider the extension F which agrees with f on S, and is constant along the lines normal to S. Then the Laplacian of f is obtained by restrict- ing AF to the original surface S. In particular, when S is the unit sphere we ob- tain the Laplacian As as follows.

DEFINITION 1. Iff is a function on Sn- 1, then Eof(x) =f(x/1 x| ) for x# O. If g is defined in a neighborhood of Sn-1, Rg is its restriction to Sn-1. Then Asf -RAEof.

A function F is homogeneous of degree k if F(tx) = tkF(x) for each t>0; this agrees with the notion of homogeneity of a polynomial. It is easy to show that if F is homogeneous of degree k, then 9F/9x1 is homogeneous of degree k-1. In particular, in Definition 1, Eof is homogeneous of degree zero, and so AEof is homogeneous of degree -2. We exploit this in the proof of Lemma 1.

LEMMA 1. f n-IfAsg =fsfn-lgAsf.

Proof. Let F Eof and G = Eog. Using the homogeneity of F and G, and the

115



116 PAPERS IN ANALYSIS

fact that the volume element dx is r'l-ldo (with r x |x and do the volunme element on Sn-1), we have

frn-3 dr f (gAsf - fAsg) do= (GAF -FAG) dx 1 ,,Sa-1 1 ~~~~~~~~~~~~~~~~~~~< I xl C 2

-= f (GOFIOr - FOGIOr) - f (GOFIOr - FOG/Or), lX_2 Il=:l

by G;reen's theorem. Since F and G are homiogeineous of degree zero, OFI Or and aGlar vanish and the lenmma followAs.

LE_MMA 2. IfJ F is homogeneouls of degree k, i.e. F(x) x I 'If( xl ) xfor some f on Sn-I, then AF(x)=k(k+n-2) I X k-2f9(XI I |)+ f XI |k--2Asf(xI I XI).

The proof is a straightforwTard verification, but may be eased by a fewA gen- eralities. First, A(HG) :HAG+2VH VG+GAH; we apply this with H(x)

x xI-, and G=Evf. Second, with this H and G, VH is perpendicular to the sphere I x constant, and VG is tangential to that sphere, so VIP VG = 0. Since AEof is homiogeneotus of degree -2, | X I k-2ASf(XI I X | ) = I X I kAEof , and it remains only to show that A( I XI ) =h k(k+-n-2) 1 xI k-2, We leave this to the reader, and consider Lemma 2 established.

REMAARK. For a function F homiogeneous of degree k,

k(k + t - 2)r-2F + r-2AsF = r1',(O Or)(rn 1OF1Or) + r-2AsF,

from which it folloNws that this last expression (being independent of k) is the polar coordinate form of A. Thus As is the "angular" part of the expression for A in polar coordinates, and e.g. when n =1, Al 2/ 062, and when n=2

1 07 OF\ 1 02F ASF == -- - sin4 - . +

sin O I0 _ do sin 2 0 O02

We do not rely, however, on these explicit expressions for As.

DEFINITION 2. Pk denotes the space of polynomials of degree < k, and P' those homogeneouts of degree k. HI- denotes the space of harmonic polynomials homogeneouts of degree k, that is the soltutions p in Pk of Ap = O.

LEMMA 3. If p P" and fs5nIpq =0 for all q in PZl2, then p is harmonic. Proof. If qCP`, wAre see by Lemmas 1 and 2 that

f qAp = k(k + n -2) f pq + qASp = qAsp pAsq

- p[r2Aq - (k - 2)(k + 4-4)q] = 0,



SPHERICAL HARMONICS 11 7

all integrals being over SI-1; the second and fifth equatlities hold since q and r2AqEP`j2. Lemmiia 3 follows w-hen we take for q the complex con jugate of AP.

DEFINITION 3. SHK, the space of spherical harmonics of degree k, is R(H'k), the restrictions to S-' of the harmonic polynomials of degree k.

THEOREMI 1. (a) SHk is a space oJ eigenfitnctions oJf As with eigenvaitc -k(k+n-2). (b) SHJ and SH k are orthogonal with respect to the inner proditct (p q) = fs-Pp, when j#k. (c) R(Pk) is the direct sutm /C=o SH'.

ProoJ. If p C Hk, then \e have froml Lemilma 2 that, oni SI'-', 0 = AP =k(n+k-2)Rp+AsRp, -which proves (a). Then (b) follows iimmiiiediately; for by Lemma 1 we have

0 = (Asp, q) - (p, Asq) = [-j(j + t - 2) + k(k + i - 2)](p, 1).

Then for (c) it suffices to show\\ R(Pk) C Z1=o SH1. This is clear for k = 0, since P?= HO. In general, we have

R(Pk) = R(PLl) + R 4p: p Pi, and f p = 0 for all q in Pk}

which belongs to R(Pk-1) +R(Hk) by Leiimia 3. Thus (c) is proved by induction.

THEOREM 2. dim(SHk) = (2k+n-2)(n+k-3)!/k!(n-2)! =O(k ?-2

Proof. Let dk,, denote the dimension of P'. Separating the mononlials of degree k in (xi, , x. ) x into those divisible by x,, and those not divisible by x,,, we get dk, ,=dk-l,l n+dk,n1 , and clearly dk,1 = 1, and do,= 1. Thus by induction on k+n, dk,,= (n+k-t)!/k!(n-t)!.

Now it is easy to see that R(Pk) is the direct sumll of R(P7) and R(PI-'); for if p Pi then p is odd or even \ith j, so R(Pk)GnR(Pj-') =0; and R(p) =R(r2p), with r(x) = x . Further, R is an isomorphism on P7 (by homogeneity). Finally, fromi- Theorem t(c), we get

dim(SHk) = dim R(Pk) - dim R(Pk-l) = dk,n + dI1_1, - (dk-1,I + dk_2,n),

which proves Theorem 2.

THEOREM 3. (FUNK-HECKE THEOREM). If f1 FI (t) (1 -t2) (,,-3I)2dt < x, and S,1,StSHl, then

F((KT, 'q))Sm(.i) du = S..) ISI-2 Can(1)-1 F(t)Cmd(t)(1 - t2) (-3)/2 dt, ns-1 -1

where C,,(t) is the Gegenbatner polynomial CxG(t) with X = 2n - 1. (The f act that C,, is a Gegenbauer polynomial will not be used in this paper, but it is useful in evaluating integrals.)



11 8 PAPERS IN ANALYSIS

Proof. Let S3, , S7t, be an orthonormal basis of SH, n, and set f(o-, ) - ZS.2l(of)SitQ). Then fm is independent of the choice of SJ2; if T, is another basis, T7t, = g/;JsJ for a unitary mnatrix (gh.2), anud Z 7-tn(of) Ttg (a) =fm (o, q). Con- sequientlyf,,(Oo-, Orq) =fm(fo-, q) for any orthogonal transformation 0 of J1. I--lence f7(o, q) depends only on (o, 77); for if (o-, -) K (a', ), there is a rotation 0 such that O= and O=Jo. We define C... by CG(~(o, -7)) =f,,(a, 7). Since the S, , may be taken to be real, it follows that C... is real.

The Ftunk-Hecke theoremi depeinds on the properties of C, w -hich are established below-.

First, fronm the definition of f,,,(, -q), anld the fact that SH", is orthogonal to SHJ w-hen k#rn, we have

(1) sm Q(7) = ?n Ck((07, -q))Sm(o-) do,

when S,, CSHm; here b is the Kronecker 3. Second, w-e show that Cm, is a polynomiiial of degree m. To see this, let

a =(ai, , a) be an nt-tuple of integers ?0, and a = ai , =x axl

XCn. Since I x | m(xil | I ) is a harmonic polynomial, we have I x I...C", (xi/ (X x ) - Z1a?rm a,,x= P(x). Setting x =(cos 0, sin 0, 0, 0, 0) and X (cos 0, -sin 0, 0,

0) wAe get C',,cos 0) =P(x) =P(), so

1 1 C,(cos 0) [ P(X) + P()I 1 E a.(cos 0>)a[(sin O)a2 + (-sin 0)a2j.

2 2 | c I =IIz

Since (sin 0)a2+(-sin 0)a2 vanishes when a2 iS odd, it is a polynomlial of degree <a2 in cos 0, and so Cm(cos 0) is a polynomial in cos 0 of degree ?m.

In fact, Cm is a Gegenbauer polynomial. For fixed q, C,,((o-, 7)) ESH'M =R(Hm); thus |x| mCm(Xi lx) is a harmonic polynom-iial. WAriting out (| x IC,(,(x1l x )) =0 leads to

(1 - t2)C"$(t) + (1 -u)tC'f 1) + m(m + it - 2)C,,(t) = 0,

which is the equation of the Gegenbauer polynomial CG (t) with A = n- . (See [ 1].) Since this differential equation is singular at t= ? 1, Cm(t) = const* GC (t), since Cm is a polynomial.

Third, the Ct,, are orthogonal on [-l , l1 with respect to the weight (1 -t2)(n-3)'2dt. This may be derived from their differential equation, but we may also observe that when q is a fixed point in SI-', C1n((, q)) and Ck((-, 77)) are spherical harmonics of degrees m and k respectively so that by Theorem 1 (b),

(2) Cm(KoY, 77))Ck (KO- q)) do- S t-2 Cm (t)Ck()(1 - 12) (nt-3)/2 di Sn-1 -1

=0 when m # k.

Now from (1), no Ck vanishes identically; and by the orthogonality just



SPHERICAL HARMONICS 119

shown, { Co, ,? C } are linearly independent. Since Ck has degree < k, it follows by induction on m that Cm has degree m exactly.

Fourth, from formula 1 and the fact that for fixed q, Cm(( , 77)) CSHm, we have

Cm(M) = CM((q, 7)) f Cm((Gi, 7))Cm((<, '1)) do-

(3) Sn 2 J Cm(t)2(1 - t2)(n 3)/2 dt.

Now the theorem follows easily. First, the condition f. I F(t) ( 1 t2) (--3)/2dt < o implies there is a sequence Fj of functions continuous on [-1, I1 such that

F( -(t F(t)I (1 - t2)(-3'2d- o,

and hence also fsn-I FK, )) - F((, 7)) |S (a) do-O; so we. may take F continuous. Since f1 (I -t2) (n-3)2dt < oo, we may in the same way replace F by a polynomial approximating F uniformly. Supposing F is a polynomial of degree k, we have (since Cj is a polynomial of degree j) that F=-L ajC. From the orthogonality relation (2) and the normalization (3), we have

a | Sn-2 1 C,(1)-1 FJ F(t)Cj(t) (1 - t2) (n-3) 12 dt.

Then the theorem follows immediately from (1).

COROLLARY. If Sm SHm, then Sm(i) 1 2? < Shm Sn-Ifs-' SmI2 where h-=dim(SHm) (2m +n-2) (n +m-3) !/m!(n-2)!. Equality holds when SinCm((n ?7)).

Proof. By (1), | SmQq) 2?fCm(Qr n)2dofI Sm| 12 Since fCmG((U, ))2do is independent of -, we have

Cm ((U a,2 do,= Sn -1 II cm((Cyalq,2 dcrd

i s 1F E fJ f SM(9)Sm()Sm(o)Sm(q) drdq

- Sn $-1 11

THEOREM 4. (a) There are constants Bk, sicih that

f-1| Dap| 2< (Bia,n)21M2Ia fs,-1|p 2 for every p in Hm.



120 PAPERS IN ANALYSIS

(b) There are constants Ck,n sitch that

DaSm(x l ) X 2 < (Ca 2 ,Y)221a1- 2+n SY 2 for S,, in SH..;

here Da=ajaj (Ox1)a. (Ox,)ja,, I al = a,+ * +a,.

Proof. Clearly B0,= 1. Suppose that Bk,n has been found, aind let I a k + 1, f31 = k, and fj3-?a, Then Dap is one of the components of the gradient VD/3p

- (OD-plOx1, , aD-plOx?). We know, since p is harmonic, that

0 = f (ADOp)(DOfi) - f VDflp, VD"fp) + (Dfip)a(Dgp) 0r. Ix I_ 1 |XI 1 sn -1

Since KVDOp, VDOp) is honmogeneous of degree 2m-2k-2 and O(DOp) Or -(mr-k) r-1D-Op, we have

fSnl Dap 2 < f KVD'p, VD3p) = (m - k)(n + 2m -2k - 2) f D3p 12 sn -1 sn-I 1, Sn-1

? (m - k) (n + 2m -2k- 2) (Bk, n)2m2kf p 12

which provides Bk+l?,, proving (a). From part (a) and the Corollary of Theorem 3, we see, for p in Hrn and

x =1, that

1 (Dap) (x) 2 < m[lal Sn-1 -1 Dap 2

< (BIaI,n)2M2IaIh_IcarI Sn-1 -1 I p 2 < (B' [,n)2ilnaI??2 j2l p 2;

hm-lal is estimated in Theorem 2. Finally, if SmfClSHm we have S,,(xl xl) = x -mp(x) for a p in Hm, and so

DaSm(xl I x I) ? (a!flO!y!) Dflr-m Dyp )3+,y=a

< (a! !y!)C1A1mlIBl ,[, lmIz[?nI21 (Ji p 1 2)1/2

< Ca1,n1l1aI-1+n/2 (f S. 2)

Here we have used Leibniz's formula Da(fg)= Z3?+,=a(a!!/3!'y!)D'3fD-g. Finally, as an application of Theorem 4, we give (see [2])



A CHARACTERIZATION OF THE EXPONENTIAL SERIES 121

THEOREM 5. Letf be integrable on Sn ,let { Sm,} (m = O, 1, ;k=O, 1, hm) be an orthonormal set of spherical harmonics, Sm of degree k, and hm = dim (SHin). Let a1k =fS-l1fSm. Then there is a C? function g, g =f a.e., if and only if amk

O(m-r) for every r.

Proof. If g exists, then

amk = mr(m + n - 2)- j g(As)rSk = m (m + n - 2M f ( Sg)M

-r -r rJ . 12)/ ?M (m +n 2) (f4g~)

Conversely, if amk 0 (m-r) for every r, then by Theorems 2 and 4, EamkSt5 converges to a CGi function g. Since fsn-(g-f)Sm=O for all m and k, we have from Theorem 1(c) that fs5-(g-f)p = O for every polynomial p. Standard density results yield f-g = 0 a.e.

The preparation of this paper was supported in part by an NSF grant at Brandeis University.

References

1. A. Erdelyi, ed., Higher Transcendental Functions, vol. 2, New York, 1953. 2. A. P. Calder6n and A. Zygmund, Singular integral operators and differential equations,

Amer. J. Math., 79 (1957) 901-921.

A CHARACTERIZATION OF THE EXPONENTIAL SERIES

J. D. BUCKHOLTZ, University of Kentucky

For all sufficiently large positive integers n, K. S. K. Iyengar [4] has shown that the disc z z _ ne-2 contains no zero of E'=0 zP/p!, the nth partial sum of the power series for ez. The main purpose of this note is to show that this fact characterizes the exponential series.

THEOREM 1. Suppose that EP=0 apzP is a power series. The following two state- ments are equivalent:

(i) There is a positive number c such that for every positive integer n, Lp=0 avpz has no zero in the disc I z < nc,

(ii) anO , and 5p= a,zP is the power series for aO exp (alz/a o).

Proof. It follows from [4] that (ii) implies (i). Suppose now that (i) is true. Then

1) Xt <S I a o/an



r. t. seeley - spherical harmonics

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