function spaces, wavelets and representation theory
TRANSCRIPT
Louisiana State UniversityLSU Digital Commons
LSU Doctoral Dissertations Graduate School
2009
Function Spaces, Wavelets and RepresentationTheoryJens Gerlach ChristensenLouisiana State University and Agricultural and Mechanical College, [email protected]
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FUNCTION SPACES, WAVELETS AND REPRESENTATION THEORY
A Dissertation
Submitted to the Graduate Faculty of theLouisiana State University and
Agricultural and Mechanical Collegein partial fulfillment of the
requirements for the degree ofDoctor of Philosophy
in
The Department of Mathematics
byJens Gerlach Christensen
B.Sc. in Mathematics, University of Copenhagen, Denmark, 2003M.Sc. in Mathematics, University of Copenhagen, Denmark, 2003
M.S. in Mathematics, Louisiana State University, USA, 2005August 2009
Acknowledgements
Before we commence I would like to thank my advisor Prof. Gestur Olafsson for his guidanceduring my years of graduate study at Louisiana State University. In particular I am thankfulfor the subject he suggested for my thesis and for motivating me to attend conferences. Iwish to thank Sigurdur Helgason for hosting me at MIT for the spring semester 2008. Thegraduate school and the department of mathematics at Louisiana State University havesupported me several times for travel to conferences, for which I am grateful. I would alsolike express my grattitude to professors Robert Perlis, Jorge Morales and Jurgen Hurrelbrinkfor partially supporting me on a research stay in Vienna through the Louisiana Board ofRegents grant LEQSF(2005-00-7)-ENH-TR21. Further I have been supported for traveland summer research by NSF grants DMS-0801010 and Louisiana Board of Regents grantLEQSF(2008-10)-ENH-TR13. I am very grateful for all the support I have received whileat Louisiana State University. It was a great opportunity for me to come here and I havethoroughly enjoyed it.
I dedicate this dissertation to my parents Kirsten and Flemming Gerlach Christensen.
ii
Table of Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Theory of Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Wavelets and Representation Theory . . . . . . . . . . . . . . . . . . 3
Chapter 2 Sampling on Some Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . 72.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Bases and Frames in Hilbert Spaces . . . . . . . . . . . . . . . 72.1.2 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Band-Limited Functions . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 The Whittaker-Shannon-Kotel’nikov Sampling Theorem . . . 112.2.2 Irregular Sampling . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Short Time Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 182.3.1 Regular Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Irregular Sampling . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Common Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 3 Groups and Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 Square Integrable Representation . . . . . . . . . . . . . . . . . . . . 303.2 Feichtinger and Grochenig Theory . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Quasi Banach Spaces of Functions . . . . . . . . . . . . . . . . 353.2.2 Coorbit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.3 Discretization and Sampling . . . . . . . . . . . . . . . . . . . 373.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Coorbit Spaces for Dual Pairs . . . . . . . . . . . . . . . . . . . . . . 423.4 Lie Groups and Smooth Representations . . . . . . . . . . . . . . . . 50
3.4.1 Smooth Square Integrable Reprensentations . . . . . . . . . . 503.4.2 Coorbits for Sobolev Spaces . . . . . . . . . . . . . . . . . . . 52
iii
Chapter 4 The Special Linear Group and Bergman Spaces . . . . . . . . . . . . . . . 544.1 SL2(R), SU(1, 1) and Discrete Series Representations . . . . . . . . . 544.2 Coorbits for Discrete Series . . . . . . . . . . . . . . . . . . . . . . . 564.3 Bergman Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Chapter 5 Besov Spaces on Light Cones . . . . . . . . . . . . . . . . . . . . . . . . . 715.1 Light Cones and Group Theory . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 Light Cones as Homogeneous Spaces . . . . . . . . . . . . . . 715.1.2 Fourier Transform on Light Cones . . . . . . . . . . . . . . . . 74
5.2 Wavelets on Light Cones . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 Littlewood-Paley Decomposition and Besov Spaces on the Light Cone 82
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
iv
Abstract
This dissertation is concerned with the interplay between the theory of Banach spaces andrepresentations of groups. The wavelet transform has proven to be a useful tool in charac-terizing and constructing Banach spaces, and we investigate a generalization of an alreadyknown technique due to H.G. Feichtinger and K. Grochenig. This generalization is presentedin Chapter 3, and in Chapters 4 and 5 we present examples of spaces which can be describedusing the theory. The first example clears up a question regarding a wavelet characteriza-tion of Bergman spaces related to a non-integrable representation. The second example is awavelet characterization of Besov spaces on the forward light cone.
v
Chapter 1
Introduction
1.1 Theory of Function Spaces
The theory of function spaces has a long and rich history of which we will point out a fewmain results. Function spaces are basically vector spaces of functions with certain properties(typically Banach or Hilbert spaces), which depend on the questions we might be asking.Given an approximation problem, we need to determine the convergence properties of thespace in which we expect to find solutions. In general a question in mathematical analysis isaccompanied by a class of functions for which the question makes sense. Thus the study offunction spaces has great importance both in applied and pure mathematics. Below we givean (incomplete) account of the history of function spaces. We conclude this section with anintroduction of the homogeneous Besov spaces which we will meet later in this thesis.
The first space of functions to be investigated extensively was probably the space ofcontinuous functions C([a, b]) on a compact interval [a, b], equipped with the uniform norm‖f‖∞ = supa≤x≤b |f(x)|. The famous Weierstrass approximation theorem states that givena function f in C([a, b]) and ε > 0 there is a polynomial P such that ‖f − P‖∞ < ε. Thisresult does not include information about how to choose P . One way to do so it due to S.N.Bernstein (see [Ach56, p.30]). Suppose [a, b] = [0, 1], and define
Bn(x) =n∑
k=0
(n
k
)f(kn
)xk(1 − x)n−k,
then ‖f − Bn‖∞ → 0 and n → ∞. Thus we can write any function as a sum of the simplerfunctions xk(1 − x)n−k an idea which is the basis for classical harmonic analysis.
In 1910 Friedrich Riesz defined the class Lp(M) of p-integrable (1 < p < ∞) functionson a Lebesgue measurable set with norm
‖f‖p =(∫
M
|f(x)|p dx)1/p
F. Riesz showed that piecewise constant functions are dense in Lp and studied the continuouslinear functionals of the spaces Lp([a, b]). He proved that the continuous linear functionals of
Lp([a, b]) can be identified with the class Lp
p−1 ([a, b]) (see [Rie10, p. 475]). The Weierstrassapproximation theorem shows that f ∈ Lp([a, b]) can be approximated in norm by the
1
Bernstein polynomials Bn for p ≥ 1. This is not the only way to approximate such functions.It turns out that f ∈ Lp([−π, π]) can indeed be approximated by the partial sum
SN (x) =k=N∑
k=−Ncn(f)einx
where cn(f) = 12π
∫ −ππ
f(x)e−inx dx. In fact for f ∈ Lp([−π, π]) it is true that
limN→∞
‖f − SN‖p = 0
when 1 < p < ∞; a result proven by S. Bochner using Riesz-Thorin interpolation (see also[Kat68, p. 50]).
It turns out that the Lp-spaces spaces are not suitable spaces in which to solve partialdifferential equations. The Sobolev space W 2
1 (M) on an open set M ⊆ Rn was introducedto solve the Laplace equation
∆u = 0 on M
u = g on the boundary of M
for a given continuous function g on the boundary of M . The space W 21 (M) consists of
square integrable functions for which the distributional derivative is again a square integrablefunction; i.e.
W 21 (M) = f ∈ L2(M)|∇f ∈ L2(M)
with norm ‖f‖2W 2
1= ‖f‖2
L2 + ‖∇f‖2L2. This space can be equipped with an inner product to
make it a Hilbert space. Banach versions W pk (M) of this type of space consist of functions
whose partial derivatives of order less than k are p-integrable. More accurately if ∂α =∂α1
∂xα11. . . ∂
αn
∂xαnn
we define
W pk (M) = f ∈ Lp(M)|∂αf ∈ Lp(M) for |α| ≤ k
with norm‖f‖W p
k=
∑
|α|≤k‖∂αf‖Lp
The book [Tar07] contains many references to the development of the theory of Sobolevspaces.
A differential operator corresponds to a polynomial in a manner specified by the Fouriertransform
F(f)(w) =1√2π
∫f(x)e−ixw dx
If P is a polynomial then the differential operator P (i ∂∂x
) can be related to the polynomialP in the following sense (for f nice enough)
F(P
(− i
∂
∂x
)f)(w) = P (w)F(f)(w)
2
Extending this principle we can define a pseudodifferential operator Dm for a Fourier multi-plier m by
Dmf = F−1(mF(f))
Dealing with such operators led to the introduction of several new function spaces. As anexample the Bessel-potential spaces
P ps = f ∈ S ′|msf ∈ Lp
with norm‖f‖P p
s= ‖Dmsf‖Lp
were introduced in [AS61] to suit the operator with multiplier ms(w) = (1 + |w|2)s/2. Othertypes of spaces were introduced where the multiplier is “cut up” into pieces. Here we onlymotivate the definition of the Besov spaces. For j ∈ Z let φi be smooth functions with com-pact support in [2−j−1/3, 2−j+1/3], such that
∑j φi = 1(0,∞). We introduce the homogeneous
Besov space Bp,qs of tempered distributions (whose Fourier transform is supported on (0,∞))
with norm
‖f‖Bp,qs
=(∑
j
2−jsq‖F−1[φjF(f)]‖Lp‖q`q)1/q
<∞
The philosophy behind these spaces is that ifDm is a pseudo-differential operator with Fouriermultiplier m bounded by a polynomial, then Dm can be discretized as follows. For w ∈[2−j−1/3, 2−j+1/3] the fraction m(w)/2−js is bounded from above and below. Let mj(w) =2jsψj(w)m(w) where ψj is a smooth functions with compact support such that ψj = 1 on[2−j−1/3, 2−j+1/3]. Then the operator Dm can be written
Dmf =∑
j
F−1(F(f)φjm) =∑
j
2−jsF−1(F(f)φjmj)
If the function m is smooth enough (see [Tri83] section 2.3.7 p. 57 for a precise statement)the functions F−1mj are integrable with uniform bound and the operator Dm is boundedon Bp,q
s . Besov spaces have, among other things, been applied to spline approximations in[HV71]. The definition of the Besov spaces presented here is not the original definition by O.V.Besov. The spaces have been given equivalent characterizations in [Pee76, Tri88a, FJW91]and we have used their results to define the Besov spaces. The work by Peetre and Triebel[Pee76, Tri88a] fits nicely with wavelet theory, which was pointed out by Feichtinger andGrochenig in the 1980’s.
1.2 Wavelets and Representation Theory
The theory of the continuous wavelet transform arises from systems of functions ψa,b, whichare dilates and translates of the wavelet ψ, i.e.
ψa,b(x) =1√|a|ψ
(x− b
a
)
3
where a ∈ R∗ and b ∈ R. The wavelet transform of a function f in L2(R) is then defined tobe
Wψ(f)(a, b) =
∫
R
f(x)ψa,b(x) dx
The dilation and translation operators form a group called the (ax + b)-group and the rep-resentation
ψ 7→ ψa,b
is a unitary representation of this group on L2(R). This was pointed out by Grossmann,Morlet and Paul in the articles [GMP85, GMP86] where other representations were alsoconsidered. Further they proved that if
∫ |ψ(w)|2|w| dw <∞
then f can be reconstructed by
f =
∫
R
∫
R∗
Wψ(f)(a, b)ψa,bda db
|a|2
For applications one is interested in replacing the integral with a sum. One such discretizationcan be obtained with the Haar wavelet ψ given by
ψ =
1 x ∈ [0, 1/2),
−1 x ∈ [1/2, 1]
0 x 6∈ [0, 1]
in which case the reconstruction formula is
f =∑
j∈±Z
∑
k∈Z
Wψ(f)(2−j, 2jk)ψ2−j ,2jk
More general discretizations were achieved in, for example, [DGM86].So far we have only described wavelets in the Hilbert space setting. However, wavelets
are also of use in Banach space theory. For example it is remarkable that if ψ is smooth andrapidly decreasing and supp(F(ψ)) ⊆ (0,∞) then the norm
‖Wψ(f)‖Lp,q
s′=
((∫|Wψ(f)(a, b)|p db
)q/pas′
da)1/q
defines an equivalent norm on the Besov space Bp,qs (when s′ is chosen correctly depending
on p,q and s). This result was first stated by Feichtinger and Grochenig, as an applicationof their so-called coorbit theory [FG88, Gro91]. They characterize several classical Banachspaces by use of unitary representations of groups, and include a framework for finding seriesexpansions for these spaces. For the Besov spaces this series expansion includes the statementthat there is a Banach sequence space bp,qs such that any f ∈ Bp,q
s can be represented as asum
f =∑
n
cnψan,bn
4
for cnn ∈ bp,qs . The coefficients cnn depend continuously on f .The theory of Feichtinger and Grochenig builds on an integrable representation (π,H) of
a locally compact group G. Let Wψ(f) denote the wavelet coefficients
Wψ(f)(x) = (f, π(x)ψ)
For a submultiplicative weight m let the norm ‖f‖Lpm
be defined by
‖F‖Lpm
=
∫
G
|F (x)|pm(x) dx
The theory of Feichtinger and Grochening, in particular, requires that the representation isunitary and irreducible and that the Banach space
H1m = f ∈ H|‖f‖H1
m= ‖Wψ(f)‖L1
m<∞
is non-trivial. Further it is required that ψ is chosen such that Wψ(ψ) ∈ L1m. Denote by
(H1m)∗ the Banach space of conjugate linear continuous functionals on H1
m. Assume that thesolid banach function space Y on G (for example Y = Lpm(G)) satisfies
‖F ∗ f‖Y ≤ C‖F‖Y
for any f ∈ L1m(G). Then we can define the coorbit space
CoY = f ∈ (H1m)∗|Wψ(f) ∈ Y
with norm ‖f‖CoY = ‖Wψ(f)‖Y . Feichtinger and Grochenig show that CoY is a π-invariantBanach space which does not depend on ψ (up to norm equivalence). Subsequently theyconstruct atomic decompositions, which are shown to always exist when working with inte-grable representations. Several generalizations of the coorbit theory have been carried outin [Rau05, Rau07, FR05, DST04]. All these constructions rely on an integrability conditonand the non-triviality of spaces similar to H1
m.When studying the fundamental papers [FG88, FG89a] and [FG89b] by Feichtinger and
Grochenig, the following list of questions are natural to ask.
1. In applications the spaces H1m and (H1
m)∗ are often disregarded and replaced by asuitable Frechet space S (for example, rapidly decreasing smooth functions). Thus thecoorbit spaces are typically described by
CoY = f ∈ S∗|Wψ(f) ∈ Y
What are the conditions on the Frechet space S and the wavelet ψ which allow theconstruction of coorbits?
2. This further poses the question if/when different “reservoirs” S yield the same spaceCoY . It is interesting to see if there are natural choices for the Frechet space S. Forexample S could be the space of smooth vectors when G is a Lie group.
5
3. What is the role of the weight function m? In particular, we need to understand thisrole if we do not wish to use H1
m. Typically it is required that m(x) ≥ 1, though thisis not the case for the weights used in the characterization of the Besov spaces.
4. Is it possible to define CoY when the representation π is non-integrable? This questionwas already posed in [FG88] section 7.3 in relation to the discrete series representationof SL2(R).
5. With no integrability condition on Wψ(ψ) what is the dependence of CoY on thewavelet ψ?
6. Similar techniques have been used to describe Banach spaces of band-limited functions[FP03], in which case the representations encountered are the non-irreducible regularrepresentations. Is it possible to unify the two theories?
7. The space Y is assumed to be solid. Is it possible to use non-solid function spaces onthe group G (for example Sobolev spaces on Lie groups)?
This thesis presents work to better understand the construction by Feichtinger andGrochenig and to suggest a generalization of their theory which answers the questions statedabove. Specifically, we construct a less involved theory with precise conditions on whichFrechet spaces S and wavelets ψ yield the same coorbit space CoY . It is the hope that thiswill promote the theory and application of coorbits to a wider audience. We do not presentgeneral conditions for the construction of atomic decompositions, but show that such de-compositions are possible in specific examples. We even give an example of a non-integrablerepresentation for which atomic decompositions exist. We further carry out a wavelet char-acterization of Besov spaces on the forward light cone, which generalizes the characterizationof the Besov spaces mentioned above. The example of the Besov spaces does not shed newlight on the coorbit theory, but is interesting in its own right.
We have organized the text as follows: Chapter 2 gives Hilbert space examples of what isto come later in the thesis. In particular, we treat sampling theorems for band-limited func-tions and for the short time Fourier transform. We then investigate the common features ofthese two subjects. In Chapter 3 we present the theory of Feichtinger and Grochenig withoutproofs and then propose a generalization of their work. In particular section 3.3 contains theanswers to question 1,2 and 5 (cf. Theorems 3.19, 3.22 and 3.21 respectively). Question 7is answered in section 3.4.2 where we give examples of coorbits related to Sobolev spaces onLie groups. The role of the weight function is determined in the sense that it does not showup in the formulation of our generalized coorbit theory. The generalization also replaces theirreducibility of the representation by a cyclicity condition, thus answering question 6 (theHilbert space case in section 2.2 illustrates this point). The last two chapters are devotedto examples. Chapter 4 treats Bergman spaces and answers question 4 by construction ofa coorbit space for a non-integrable representation. In section 4.4 we show, among otherthings, that it is still possible to find atomic decompositions in the non-integrable case. Thelast chapter presents a wavelet characterization of Besov spaces on the forward light cone.This has, to the author’s knowledge, not been carried out before.
6
Chapter 2
Sampling on Some Hilbert Spaces
In this chapter we investigate sampling results in certain Hilbert spaces. We cover the famousWhittaker-Shannon-Kotel’nikov sampling theorem for band-limited functions, which statesthat a function can be reconstructed by knowing its values at certain equidistant points(regular sampling). We further will present results for irregular sampling of band-limitedfunctions. Next we turn to both regular and irregular sampling of the short time Fouriertransform. It is worth noting that the results on irregular sampling for the band-limitedfunctions is actually used to carry out irregular sampling the short time Fourier transform.We next show how groups and group representations play a role in the structure of the spaceswe sample. The group theory gives a tool for obtaining an isomorphism between the originalHilbert space and a reproducing kernel Hilbert space of functions. The aim of this chapteris to motivate a similar theory for Banach spaces and to illuminate the underlying structureof two seemingly different areas of harmonic analysis.
2.1 Prerequisites
Before we state the results of this chapter we introduce the notion of frames on Hilbertspaces and the Fourier transform on L2(Rn).
2.1.1 Bases and Frames in Hilbert Spaces
In the following let H be a separable Hilbert space with inner product (·, ·) and norm ‖ · ‖ =√(·, ·). If T : H → H is a continuous operator onH the operator norm ‖T‖ = sup‖f‖=1 ‖Tf‖
is bounded. If ‖I−T‖ < 1 then the operator T is invertible with inverse given by its Neumannseries
T−1f =
∞∑
k=0
(I − T )kf
A system eii∈I is called an orthonormal basis if (ei, ej) = δi,j (δi,j is the Kronecker delta)and the Hilbert space closure of
∑i∈I′ aiei|I ′ if finite is dense in H . For an orthonormal
basis Parseval’s equality
‖f‖2 =∑
i∈I|(f, ei)|2
7
holds for all f ∈ H and each f ∈ H can be written uniquely as
f =∑
i∈I(f, ei)ei
where the sum is regarded as a H-limit of partial sums. If we transmit information bysending the coefficients (f, ei) we will be able to reconstruct f completely. However if wehave a faulty transmission, we would lose vital information every time a coefficient is nottransmitted. Thus it can be useful to work with systems of higher redundancy. Yet wewould like to still be able to reconstruct using a series expansion. Frames were introduced in[DS52], and provide us with the appropriate framework for this purpose. A sequence ψii∈Iis called a frame for the Hilbert space H if there are constants 0 < A ≤ B <∞ such that
A‖f‖2 ≤∑
i∈I|(f, ψi)|2 ≤ B‖f‖2
In that case the operator
Sf =2
A +B
∑
i∈I(f, ψi)ψi
is a bounded positive operator on H . By the calculation
(f − Sf, f) = ‖f‖2 − (Sf, f) ≤(1 − 2A
A+B
)‖f‖2 =
B −A
A +B‖f‖2 < ‖f‖2
we see that ‖I − S‖ ≤ B−AA+B
< 1 and then S is invertible with
S−1 =∑
k
(I − S)k
The fact that S is invertible, provides us with the series expansion (using that S is positive)
f =∑
i∈I(f, ψi)ψi
and a similar expansion
f =∑
i∈I(f, ψi)ψi
where the elements ψi = S−1ψi are called the dual frame of ψi. The problem is now how toinvert the operator S. This is particularily easy when the framebounds A and B are equal,in which case the frame is called tight and the operator S is the identity operator. If theframe is not tight we can use the Neumann series to obtain an inversion algorith
Theorem 2.1 (Frame Algorithm). It is possible to approximate f ∈ H using the values(f, ψk) in the following manner
f0 = Sf
fn =
n∑
k=0
(I − S)kf0 = (I − S)fn−1 + f0
‖f − fn‖H =∥∥∥
∞∑
k=n+1
(I − S)f0
∥∥∥ ≤ ‖I − S‖n+1‖f‖ ≤(B −A
A+B
)n+1
‖f‖
8
We see that the smaller the constant B−AA+B
is the faster this algorithm converges. Thusit is desirable to obtain frame bounds A and B as close to each other as possible, with theoptimal situation being a tight frame.
2.1.2 Fourier Transforms
In the following let L2(R) denote the Hilbert space of square integrable functions on [−π, π]with respect to Lebesgue measure. The inner product on this spaces is given by
(f, g)L2 =
∫f(x)g(x) dx
and the induced norm denoted by ‖f‖2 =√
(f, f)2. Further the space `2(Z) denotes thespaces of square summable sequences with inner product
(x, y)`2 =∑
n∈Z
xnyn
and induced norm ‖x‖`2 =√
(x, x)`2 . The discrete Fourier transform of f ∈ L2([−π, π]) isdefined as
f(n) =1√2π
∫ π
−πf(x)en(x) dx
where en(x) = 1√2πeinx. The sequence f(n)n∈Z is in `2(Z) and Parsevals equality states
that‖f‖2
`2 = ‖f‖2L2
which shows that the discrete Fourier transform is an isometry from L2([−π, π]) to `2(Z). Thefunctions en form an orthonormal basis for L2([−π, π]) and any function can be reconstructedby
f =∑
n∈Z
f(n)en
with L2-convergence of the sum.For L2(Rn) there is a generalization of the Fourier transform, which we will present here.
Let S(Rn) denote the space of rapidly decreasing smooth functions on Rn with semi norms
‖f‖k,l = sup|α|≤k,
‖(1 + |x|2)l∂αf(x)‖∞
For f ∈ S(Rn) (or f ∈ L1(Rn)) define the Fourier transform
Ff(w) = f(w) =1√2π
n
∫
R
f(x)e−i(x,w) dx
It is well known that f ∈ S(Rn) (or f is continuous and bounded when f ∈ L1(Rn)) andthat the Fourier transform can be inverted by
f(x) =1√2π
n
∫
R
f(w)ei(x,w) dw = F(f)(−x)
9
Further we know that‖f‖L2 = ‖f‖L2
and since the rapidly decreasing functions are dense in L2 the mapping F : S(Rn) → S(Rn)
extends to a unitary isomorphism F : L2(Rn) → L2(Rn) (we will also use f for Ff whenf ∈ L2(Rn)). For the Fourier transform holds that F2f(x) = f(−x) and thus also F4f(x) =f(x).
When the functions f, g are in L2(Rn) the convolution f ∗ g is well-defined by the expres-sion
f ∗ g(x) =
∫f(y)g(x− y) dy
and if f ∗ g ∈ L2(Rn) then
F(f ∗ g) =√
2πnF(f)F(g)
In section 2.2 we will use the discrete Fourier transform on any symmetric interval [−Ω,Ω],so we finish this section by listing some of its properties. The Fourier transform on [−Ω,Ω] isobtained by dilating functions and applying the known results for the Fourier transform onthe interval [−π, π]. If g ∈ L2([−Ω,Ω]) then the function gΩ(x) = g(Ωx/π) is in L2([−π, π])and
‖gΩ‖22 =
∫ π
−π|gΩ(x)|2 dx =
π
Ω
∫ Ω
−Ω
|g(x)|2 dx = ‖g‖22
If further g ∈ L1([−Ω,Ω]) then
gΩ(n) =π
Ωg(nπ
Ω
)
Therefore the Parseval equality for g ∈ L2([−Ω,Ω]) becomes
‖g‖22 =
π
Ω
∑
n
∣∣∣g(nπ
Ω
)∣∣∣2
and en(x) = 1√2Ωein
πΩx forms an orthonormal basis for L2([−Ω,Ω]).
2.2 Band-Limited Functions
In this section we will work with functions f for which the Fourier transform f has compactsupport in the interval [−Ω,Ω]. We call these functions Ω-band-limited and denote the classof such functions by
L2Ω(R) = f ∈ L2(R)|supp(f) ⊆ [−Ω,Ω]
Let ψ ∈ L2Ω(R) be the function with Fourier transform ψ = 1√
2π1[−Ω,Ω] ∈ L1(R). Then ψ has
a continuous representative in L2(R) and in fact we can choose
ψ(x) =Ω
πsinc(Ωx)
where
sinc(x) =
sinxx
x 6= 0
1 x = 0
Since ψ is bounded the convolution f ∗ ψ has the following properties
10
1. f ∗ ψ ∈ L2(R) for f ∈ L2(R) and ‖f ∗ ψ‖L2 ≤ ‖f‖L2
2. ‖f ∗ ψ‖L2 = ‖f‖L2 for f ∈ L2Ω(R)
This shows that f 7→ f ∗ ψ is a projection from L2 to L2Ω(R).
Also note that since supp(f) ⊆ [−Ω,Ω] the function
f(z) =1√2π
∫ Ω
−Ω
f(w)eiwz dw =1√2π
∫ Ω
−Ω
f(w)eiwRe(z)e−wIm(z) dw
is a holomorphic extension of f with growth |f(z)| ≤ ‖f‖L1√2πeΩ|Im(z)|. Therefore a function
f ∈ L2Ω(R) is determined by its values on a set with a limit point. When dealing with
unbounded sets the situation is more complicated, yet in some cases it is possible to giveexact reconstruction formulas. We will investigate this in the cases where the points xn areeither evenly spaced (regular sampling) or unevenly spread with a fixed maximum distance(irregular sampling).
2.2.1 The Whittaker-Shannon-Kotel’nikov Sampling Theorem
For a function f ∈ L2Ω its Fourier transform is in L2([−Ω,Ω]). The space L2([−Ω,Ω]) has an
orthonormal basis enn∈Z given by en(y) = 1√2Ωe−i
πΩny (we have swapped the order of the
n’s on purpose) and therefore
f(x) =∑
n∈Z
(f , en)en(x)
=1
2Ω
∑
n∈Z
(∫ Ω
−Ω
f(y)eiπΩnydy
)e−i
πΩnx
=
√2π
2Ω
∑
n∈Z
f(−nπ
Ω
)e−i
πΩnx
=
√2π
2Ω
∑
n∈Z
f(nπ
Ω
)e−i
πΩnx
The sum converges in L2([−Ω,Ω]) in the following sense: if SN denotes the partial sum
SN(x) =
√2π
2Ω
∑
|n|≤Nf(nπ
Ω
)e−i
πΩnx
then ‖f − SN‖L2 → 0. The inverse Fourier transform gives
F−1(SN)(x) =1
2Ω
∑
|n|≤Nf(nπ
Ω
) ∫ Ω
−Ω
e−iπΩnyeixy dy
=1
2Ω
∑
|n|≤Nf(nπ
Ω
)(ei(xΩ−nπ) − e−i(xΩ−nπ)
i(x− πΩn)
)
=∑
|n|≤Nf(nπ
Ω
)sin(xΩ − nπ)
xΩ − nπ
11
and this partial sum converges to f in L2(R). We have just proven what is known as theWhittaker-Shannon-Kotel’nikov sampling theorem which can be found in [Whi15, Kot01,Sha49].
Theorem 2.2. If f ∈ L2Ω(R) then f can be recovered from its samples
f(nπΩ
)
n∈Z
in the
following manner
f(x) =∑
n∈Z
f(nπ
Ω
)sin(xΩ − nπ)
Ωx− nπ
with convergence of partial sums in L2(R).
2.2.2 Irregular Sampling
We will now look at results concerning irregular sampling of band limited functions. Butfirst we introduce some notation. By 1A we denote the indicator function of a measurableset A. We also introduce the local oscillations for f depending on a neighbourhood U by
oscU(f)(x) = supu∈U
|f(x) − f(x+ u)|
The first result is proved using techniques similar to Theorem 5 from [Gro92].
Theorem 2.3. Suppose that f ∈ L2Ω(R). If (xn) is any increasing sequence such that δ
satisfies
δ = sup(xn+1 − xn) <
√2
Ω
and also limn→±∞ xn = ±∞ then
(1 − δΩ√
2
)2
‖f‖22 ≤
∑
n∈Z
xn+1 − xn−1
2|f(xn)|2 ≤
(1 +
δΩ√2
)2
‖f‖22
and the functions ψn(x) =√
xn+1−xn−1
2ψ(x−xn) form a frame for L2
Ω. If ψn denotes the dual
frame then f can be reconstructed from its samples by
f =∑
n
√xn+1 − xn−1
2f(xi)ψn
Proof. First let yn = (xn + xn+1)/2 be the midpoints of the intervals [xn, xn+1] and notethat the intervals Un = [yn−1, yn] are pairwise disjoint and contained in xn + U where U =[−δ/2, δ/2]. Then look at
∥∥∥f −∑
n
f(xn)1Un
∥∥∥2
2=
∥∥∥∑
n
(f − f(xn))1Un
∥∥∥2
2
=∑
n
∫ yn
yn−1
|f(x) − f(xn)|2 dx
12
When x ∈ Un ⊆ xn + U then xn ∈ x + U , so |f(x) − f(xn)| can be estimated by the localoscillations and
∥∥∥f −∑
n
f(xn)1Un
∥∥∥2
2≤
∑
n
∫ yn
yn−1
(oscU(f)(x))2 dx = ‖oscU(f)(x)‖22
Since f is differentiable (f has a holomorphic extension) we get that
(oscU(f)(x))2 = supu∈U
|f(x) − f(x+ u)|2
= supu∈U
∣∣∣∫ x+u
x
f ′(t) dt∣∣∣2
≤ supu∈U
( ∫ x+u
x
|f ′(t)| dt)2
≤ supu∈U
∫ x+u
x
1 dt
∫ x+u
x
|f ′(t)|2 dt
≤ supu∈U
|u|∫ x+u
x
|f ′(t)|2 dt
≤ δ
2max
∫ x+δ/2
x
|f ′(t)|2 dt,∫ x
x−δ/2|f ′(t)|2 dt
≤ δ
2
∫ x+δ/2
x−δ/2|f ′(t)|2 dt
=δ
2
∫ δ/2
−δ/2|f ′(t+ x)|2 dt
Integrating we then get
‖oscU(f)‖22 ≤
δ
2
∫
R
∫ δ/2
−δ/2|f ′(t+ x)|2 dt dx
=δ
2
∫ δ/2
−δ/2
∫
R
|f ′(t+ x)|2 dx dt
=δ2
2‖f ′‖2
2
We now use that f is band-limited to estimate
∫
R
|f ′(x)|2 dx =
∫ Ω
−Ω
|wf(w)|2 dw ≤ Ω2
∫ Ω
−Ω
|f(w)|2 dw = Ω2
∫
R
|f(x)|2 dx
Therefore we obtain ∥∥∥f −∑
n
f(xn)1Un
∥∥∥2
2≤ (δΩ)2
2‖f‖2
2
Noting now that ∣∣∣‖f‖2 −∥∥∥
∑
n
f(xn)1Un
∥∥∥2
∣∣∣ ≤∥∥∥f −
∑
n
f(xn)1Un
∥∥∥2
13
and ∥∥∥∑
n
f(xn)1Un
∥∥∥2
2=
∑
n
|f(xn)|2∫ yn
yn−1
1 dx =∑
n
xn+1 − xn−1
2|f(xn)|2
we get the result
(1 − δΩ√
2
)2
‖f‖22 ≤
∑
n
xn+1 − xn−1
2|f(xn)|2 ≤
(1 +
δΩ√2
)2
‖f‖22
This inequality does not immediately look like a frame inequality. But if we note that
f(xn) = f ∗ ψ(xn) =
∫
R
f(x)ψ(xn − x) dx =
∫
R
f(x)ψ(x− xn) dx = (f, ψ(· − xn))
then the inequality produces
(1 − δΩ
2√
2
)2
‖f‖22 ≤
∑
n
|(f, ψn)|2 ≤(1 +
δΩ
2√
2
)2
‖f‖22
to show that ψn(x) =√
xn+1−xn−1
2ψ(x − xn) is a frame. The reconstruction formula follows
directly from the same calculation and the definition of the dual frame.
A better result, which allows the sampling points to be a larger distance apart, can beobtained if we refrain from using the local oscillation oscU(f). The following can be foundin [Gro92] as part of Theorem 1 and as Theorem 4 in [Gro93] with a slightly better resultthan presented here.
Theorem 2.4. Suppose that f ∈ L2Ω(R). If (xn) is any increasing sequence such that δ
satisfies
δ = sup(xn+1 − xn) <2√
2
Ω
and also limn→±∞ xn = ±∞ then
(1 − δΩ
2√
2
)2
‖f‖22 ≤
∑ xn+1 − xn−1
2|f(xn)|2 ≤
(1 +
δΩ
2√
2
)2
‖f‖22
and the functions ψn(x) =√
xn+1−xn−1
2ψ(x−xn) form a frame for L2
Ω. If ψn denotes the dual
frame then f can be reconstructed from its samples by
f =∑
n
√xn+1 − xn−1
2f(xi)ψn
Proof. The argument is the same as before, except that we avoid the use of oscU(f) in theestimates
∫ yn
yn−1
|f(x) − f(xj)|2 dx =
∫ yn
yn−1
∣∣∣∫ x
xn
f ′(t) dt∣∣∣2
dx
14
which by Holder’s inequality can be estimated by
≤∫ yn
yn−1
∫ x
xn
|f ′(t)|2 dt∫ x
xn
1 dt dx
=
∫ xn
yn−1
(x− xn)
∫ x
xn
|f ′(t)|2 dt dx+
∫ yn
xn
(x− xn)
∫ x
xn
|f ′(t)|2 dt dx
integration by parts in x gives
=1
2
[(x− xn)
2
∫ x
xn
|f ′(t)|2 dt]xn
yn−1
− 1
2
∫ xn
yn−1
(x− xn)2|f ′(x)|2 dx
+1
2
[(x− xn)
2
∫ x
xn
|f ′(t)|2 dt]xn
yn
− 1
2
∫ yn
xn
(x− xn)2|f ′(x)|2 dx
evaluating and joining intervals yields
=1
2(yn−1 − xn)
2
∫ xn
yn−1
|f ′(t)|2 dt+1
2(yn − xn)
2
∫ yn
xn
|f ′(t)|2 dt
− 1
2
∫ yn
yn−1
(x− xn)2|f ′(x)|2 dx
≤ 1
2max(yn−1 − xn)
2, (yn − xn)2
∫ yn
yn−1
|f ′(t)|2 dt
− 1
2
∫ yn
yn−1
(x− xn)2|f ′(x)|2 dx
<δ2
8
∫ yn
yn−1
|f ′(x)|2 dx
where we in the last inequality have used that the yn’s are midpoints of intervals of lengthδ and that the last term is positive.
Remark 2.5. As mentioned [Gro93] contains an even better estimate obtain by use ofWirtinger’s inequality (see [HLP52] p. 184). In particular the frame inequality obtainedthere is (
1 − δΩ
π
)2
‖f‖22 ≤
∑
j
xj+1 − xj−1
2|f(xj)|2 ≤
(1 +
δΩ
π
)2
‖f‖22
whenδ = sup(xn+1 − xn) <
π
Ω
The sampling rate Ωπ
is the minimal rate for which we can reconstruct the signal. It alsoshowed up in the Whittaker-Shannon-Kotel’nikov sampling theorem and is referred to as theNyquist rate. Theorem 2.4 is thus not optimal, but it has been included to show how simpleproperties of band-limited functions improve estimates that were obtained by use of localoscillations ocsU(f).
15
Remark 2.6 (A different series expansion). Part of the proof of Theorem 2.4 can be ex-tracted to obtain a series expansion and a reconstruction algorithm with faster convergencethan the frame algorithm (Theorem 2.1) If f ∈ L2
Ω and xn are chosen as in Theorem 2.4then ∥∥∥f −
∑
n
f(xn)1[yn−1,yn]
∥∥∥2≤ δΩ
2√
2‖f‖2
The function∑
n f(xn)1[yn−1,yn] is not necessarily in L2Ω (the partial sums are not), so the
operator S : L2Ω → L2(R) given by
Sf =∑
n
f(xn)1[yn−1,yn]
cannot be inverted using the Neumann series. However, since convolution with ψ is acontinuous projection, the estimate above ensures that we can in fact invert the operatorT : L2
Ω → L2Ω defined by
Tf =∑
n
f(xn)1[yn−1,yn] ∗ ψ
In particular the estimate shows that for f ∈ L2Ω
∥∥∥f −∑
n
f(xn)1[yn−1,yn] ∗ ψ∥∥∥
2=
∥∥∥f ∗ ψ −∑
n
f(xn)1[yn−1,yn] ∗ ψ∥∥∥
2
≤∥∥∥f −
∑
n
f(xn)1[yn−1,yn]
∥∥∥2
≤ δΩ
2√
2‖f‖2
Therefore ‖I − T‖ ≤ δΩ2√
2and T is invertible with inverse
T−1 =∑
k
(I − T )k
Thus f can be reconstructed asf = T−1(Tf)
and an approximation algorithm can be obtained from the Neumann series:
f0 = Tf,
fN =
N∑
k=0
(I − T )kf0 = (I − T )fN−1 + f0
‖f − fN‖2 = ‖∞∑
k=N+1
(I − T )kf0‖2 = ‖(I − T )N+1∞∑
k=0
(I − T )kf0‖2
≤ ‖I − T‖N+1‖f‖2 ≤( δΩ
2√
2
)N+1
‖f‖2
16
If we let γ = δΩ2√
2< 1 this algorithm has convergence
‖f − fN‖2 ≤ γN+1‖f‖2
while the frame algorithm (Theorem 2.1) displays the slower convergence (with A = (1−γ)2
and B = (1 + γ)2)
‖f − fN‖2 ≤( 2γ
1 + γ2
)N+1
‖f‖2
This however comes with the added expense of having to convolute every 1[yn−1,yn] with ψ.The fact that T is invertible provides us with a series expansion for f in terms of the atomsψn = T−1(1[yn−1,yn] ∗ ψ) by
f =∑
n
f(xn)ψn
2.2.3 Group Theory
We now show how representation theory enters the picture. This might seem a bit forcedfor the example of band-limited functions, but we wish to point out similarities between thetheory for band-limited functions and short time Fourier transforms (see next section).
The left translation`yf(x) = f(x− y)
for f ∈ L2(R) is a unitary representation of R. Using the Fourier transform we get theequivalent representation
yf(w) =
1√2πeiywf(w)
on L2(R). From this we see, that for each Ω > 0 the space of band-limited functions L2Ω(R)
is both closed and `-invariant, so the representations ` and are not irreducible. Evenrestricting ` to L2
Ω(R) for some Ω does not give an irreducible representation, yet each suchspace has a cyclic vector (ψ is cyclic if ∀y : (f, `yψ) = 0 ⇒ f = 0). This is enough to carryout a construction similar to the classification of L2(R) which we will treat in section 2.3.
Let ψ be he function defined such that ψ = 1√2Ω
1[−Ω,Ω], then ψ is cyclic in L2Ω(R). This
follows from the fact that en(w) = 1√2Ωe−inΩ/πw forms an orthonormal basis for L2([−Ω,Ω])
and that (`−nΩ/πψ, f)L2Ω
= (en, f). For f ∈ L2(R) we define the wavelet transform
Wψ(f)(y) = (f, `yψ) =
∫
R
f(x)ψ(x− y)dx = f ∗ Ψ(y)
where Ψ(x) = ψ(−x). Notice that Ψ = ψ and therefore
Wψ(f) = f ∗ ψThe function Wψ(f) is square integrable as can be seen like this
∫
R
|Wψ(f)(y)|2dy =
∫
R
∣∣∣∫
R
f(x)ψ(x− y)dx∣∣∣2
dy
=
∫
R
∣∣∣∫
R
f(x)ψ(x)e2πixydx∣∣∣2
dy
17
The product h = f ψ is in L1(R) ∩ L2(R) and thus the expression above becomes
=
∫
R
|F−1(h)(y)|2dy
=
∫
R
|h(y)|2dy
=
∫
R
|f ψ|2dy
≤ ‖f‖2L2(R)
Equality will only hold if f ∈ L2Ω(R), and therefore Wψ is an isomorphism from L2
Ω(R) ontothe Hilbert space
HΩ = f ∈ L2Ω(R)|Wψ(f) ∈ L2(R)
with norm ‖f‖HΩ= ‖Wψ(f)‖L2(R). We also have the reproducing formula
Wψ(f) = Wψ(f) ∗Wψ(ψ)
which shows that Wψ(ψ) is a reproducing kernel for the Hilbert space HΩ, and then pointevaluation of Wψ(f) is continuous. Since Wψ(f) = f ∗ ψ = f for f ∈ L2
Ω the frame boundfrom Theorem 2.4 is also a frame bound for Wψ(f) ∈ HΩ. The mapping L2(R) 3 F 7→F ∗Wψ(ψ) ∈ HΩ is a continuous projection which is a crucial tool in order to obtain thediscretization in Remark 2.6 for Wψ(f) ∈ HΩ.
Again, these remarks are somewhat trivial in the case of band-limited functions, yet weare provided with an example where the wavelet coefficient Wψ(ψ) is not integrable. We willreturn to this later.
2.3 Short Time Fourier Transform
The Fourier transform
f(w) =1√2π
∫f(x)e−ixw dx
gives information about the frequency content of a signal f ∈ L2(R). However a musicalpiece is highly dependent on when the frequencies occur. The Fourier transform gives no suchinformation, so we need to find other techniques to analyse time and frequency resolution ofsignals. One way to gain knowledge about which frequencies occur at a certain time is byuse of the Short Time Fourier Transform or Windowed Fourier Transform
Sg(f)(t, w) =1√2π
∫f(x)g(x− t)e−ixw dx
The function g is called the window and is typically used to cut off the signal f around acertain time (see figure).
Let g∗w(x) = g(−x)eixw then g∗w(s) = g(w + s) and
Sg(f)(t, w) =1√2πf ∗ g∗w(t)
18
g(x− t)g(x− t)
f(x)f(x)
←− t −→←− t −→
f(x)f(x)
f(x)g(x− t)f(x)g(x− t)
←− t −→←− t −→
Figure 2.1: Window use in Short time Fourier transform
We can use the Fourier transform (in the t variable) to obtain
Ft(Sg(f)(·, w)) =1√2πf g∗w
By use of Fubini’s theorem we can show that Sg(f) ∈ L2(R2) when f, g ∈ L2(R) as follows
2π
∫∫
R2
|Sg(f)(t, w)|2 dt dw =
∫∫
R2
|f(s)|2|g∗w(s)|2 ds dw
=
∫∫
R2
|f(s)|2|g(w + s)|2 dw ds
= ‖f‖22‖g‖2
2
The same calculations show that f can be reconstructed (in the L2-sense) from its ShortTime Fourier Transform by (choosing ‖g‖2
2 = 12π
)
f(x) =
∫
R2
Sg(f)(t, w)g(x− t)eiwx dt dw
If we define the twisted convolution
F#G(t, w) =
∫F (s, y)G(t− s, w − y)eis(y−w) ds dy
we are also able to obtain a reproducing formula
Sg(f)#Sg(g)(t, w) = Sg(f)(t, w) (2.1)
If we assume that the mapping
L2(R2) 3 F 7→ F#Sg(g) ∈ L2(R2)
is continuous then it is a projection. This is definitely the case if Sg(g) ∈ L1(R2) and thenthe image of L2(R) under the short time Fourier transform is the reproducing kernel Hilbertspace L2(R2)#Sg(g) (we will show this later). The functions Sg(f) are continuous and we canhope to reconstruct Sg(f) from its samples. This can then in turn be used to reconstructf from the samples of Sg(f). In the following sections a few sampling theorems will beinvestigated.
19
2.3.1 Regular Sampling
Example 2.7 (Orthogonal Basis). Let g = 1√2π
1[−π,π[ and gk,j(x) = g(x − 2πk)eijx. The
short time Fourier transform of f at the points (2πk, j) is then
Sg(f)(2πk, j) =1√2π
∫
R
f(x)g(x− 2πk)e−ijxdx
=1√2π
∫
R
f(x+ 2πk)g(x)e−ij(x+2πk)dx
=e−2πijk
2π
∫ π
π
fk(x)e−2πijxdx
=1
2πfk(j)
where fk(x) = f(x+ 2πk)1[−π,π[(x). Using the discrete Fourier transform, we can obtain fkfrom its Fourier coefficients fk(j) in the following manner
fk(x) =∑
j∈Z
fk(j)eijx1[−π,π[(x) =
∑
j∈Z
2πSg(f)(2πk, j)eijx1[−π,π[(x)
with convergence in L2([−π, π[). Then finally we know that
f(x) =∑
k∈Z
fk(x− 2πk)
= 2π∑
k∈Z
∑
j∈Z
Sg(f)(2πk, j)1[−π,π[(x− 2πk)eij(x−2πk)
= 2π∑
k∈Z
∑
j∈Z
Sg(f)(2πk, j)1[−π,π[(x− 2πk)eijx
in the weak sense. This can also be written as
f = (2π)3/2∑
k∈Z
∑
j∈Z
Sg(f)(2πk, j)gk,j (2.2)
This shows that we can reconstruct an f ∈ L2(R) from the samples Sg(f)(2πk, j) with(k, j) ∈ Z2. What has really been exploited here is that gk,j forms an orthonormal basis ofL2(R).
So far the functions gk,j have been non-continuous. Cutting off signals with indicatorfunctions introduces high frequency components in the coefficients Sg(f) even if f is smooth.In order to work with continuous or smooth windows, we lose the orthogonality of thetranslates gj,k. This however is not a problem as long as we can construct a (tight) frame.This is done in Theorem 1 and Theorem 2 in the article [DGM86]. Here we present the caseof a tight frame, which allows reconstruction of f from Sg(f)(αk, βj) for k, j ∈ Z .
Theorem 2.8 (Daubechies, Grossman, Meyer). Assume that g is real and continuous withsupport in [−L,L] and bounded away from zero on [−Lµ, Lµ] where 0 < µ < 1. If
∑
m
|g(x− 2µmL)|2 = c
20
then the functions gm,n(x) = g(x − 2mµL)ei2πnx/L form a tight frame for L2(R) and f canbe reconstructed by
f =
√2π
2Ωc
∑
n,m
Sg(f)(2mµL, 2πn/L)gm,n
Proof.
∑
n
|(f, gm,n)|2 =∑
n
∣∣∣∫f(x)g(x− 2mµL)e−i2πnx/L dx
∣∣∣2
=∑
n
∣∣∣e−in2mµL
∫f(x+ 2mµL)g(x)e−i2πnx/L dx
∣∣∣2
let hm(x) = f(x+ 2mµL)g(x) which is supported in [−L,L] then
= 2π∑
n
|hm(2πn/L)|2
and Parsevals identity gives
= 2Ω
∫|hm(x)|2 dx
= 2Ω
∫|g(x)|2|f(x+ 2mµL)|2 dx
= 2Ω
∫|g(x− 2mµL)|2|f(x)|2 dx
Summing over m and using the dominated convergence theorem we obtain
∑
m
∑
n
|(f, gn,m)|2 = 2Ω
∫ ∑
m
|h(x−mµL)|2|f(x)|2 dx
= 2Ωc
∫|f(x)|2 dx
This shows that gm,n forms a tight frame. Since the frame is tight its dual is (2Ωc)−1gm,nand the reconstruction formula follows by noting that
(f, gm,n) =√
2πSg(f)(2mµL, 2πn/L)
To illustrate this theorem we provide the following example also found in [DGM86].
Example 2.9 (Regular Sampling). With L = 1 and µ = 1/2 the function
g(x) =
0 |x| ≥ 1
cos(πx/2) |x| ≤ 1
21
|g(x−m + 1)|2|g(x−m + 1)|2|g(x−m + 1)|2 |g(x−m)|2|g(x−m)|2|g(x−m)|2 |g(x−m− 1)|2|g(x−m− 1)|2|g(x−m− 1)|2
m− 1m− 1m− 1 mmm m + 1m + 1m + 1
|g(x−m + 1)|2 + |g(x−m)|2 + |g(x−m− 1)|2|g(x−m + 1)|2 + |g(x−m)|2 + |g(x−m− 1)|2|g(x−m + 1)|2 + |g(x−m)|2 + |g(x−m− 1)|2|g(x−m + 1)|2 + |g(x−m)|2 + |g(x−m− 1)|2
m − 1m − 1m − 1m − 1 mmmm m + 1m + 1m + 1m + 1
Figure 2.2: A tight frame from g(x) = cos(x)
can be used to obtain a frame using the theorem above. We see that g(x−m) and g(x− n)only overlap if n = m − 1,n = m or n = m + 1 as illustrated on the figure below. Whenm− 1 ≤ x ≤ m+ 1
|g(x−m+ 1)|2 + |g(x−m)|2 + |g(x−m− 1)|2 = 1
so∑
m |g(x+m)|2 = 1.
2.3.2 Irregular Sampling
In this section we present an irregular sampling theorem for the short time Fourier transformin the case when the window function is band-limited (in which case the function cannot becompactly supported by the Hardy uncertainty principle [Har33]. The next sampling theoremis similar to Theorem 3 in [Gro93], but the formulation and proof are slightly different. Theformulation makes use of the local oscillation of a function defined on page 12.
Theorem 2.10. Given a closed interval U containing 0 let g be a band-limited even windowfunction such that ‖g‖2 = 1 and ‖oscU(|g|2)‖1 = γ < 1. Let wj be an increasing sequencesuch that wj−wj+1 ∈ U and limj→±∞wj = ±∞. For each j let tj,k be an increasing sequence
such that δ = supj,k(tj,k+1 − tj,k) <2√
2Ω
and also limk→±∞ tj,k = ±∞. Then
1 − γ
2π
(1 − δΩ
2√
2
)2
‖f‖22 ≤
∑
k,j
(tj,k+1 − tj,k−1)(wj+1 − wj−1)
4|Sg(f)(tj,k, wj)|2
≤ 1 + γ
2π
(1 +
δΩ
2√
2
)2
‖f‖22
Proof. Let yj be the midpoints of the intervals [wj−1, wj]. Then the intervals [yk, yk+1] cover
22
R with overlap of measure 0 and since ‖Sg(f)‖L2(R2) = 12π‖f‖2 we have
∣∣∣1
2π‖f‖2
2 −∑
j
wj+1 − wj−1
2‖Sg(f)(·, wj)‖2
2
∣∣∣
=∣∣∣‖Sg(f)‖2
2 −∑
j
(yj+1 − yj)‖Sg(f)(·, wj)‖22
∣∣∣
=∣∣∣∫
‖Sg(f)(·, w)‖22 −
∑
j
‖Sg(f)(·, wj)‖221[yj ,yj+1](w) dw
∣∣∣
=∣∣∣∫ ∑
j
(‖Sg(f)(·, w)‖22 − ‖Sg(f)(·, wj)‖2
2)1[yj,yj+1](w) dw∣∣∣
≤∫ ∑
j
∣∣‖Sg(f)(·, w)‖22 − ‖Sg(f)(·, wj)‖2
2
∣∣1[yj,yj+1](w) dw
We need to estimate∣∣‖Sg(f)(·, w)‖2
2 − ‖Sg(f)(·, wj)‖22
∣∣ for w ∈ [yj, yj+1] ⊆ wk + U . To dothis first note that
‖Sg(f)(·, w)‖22 =
∫|Sg(f)(t, w)|2 dt
=1
2π
∫|∫f(x)e−iwxg(x− t) dx|2 dt
=1
2π
∫|∫f(y + w)g(y)e−ity dy|2 dt
=1
2π
∫|f(y + w)|2|g(y)|2 dy
=1
2π
∫|f(y)|2|g(y − w)|2 dy
=1
2π
∫|f(y)|2|g(w − y)|2 dy
=1
2π|f |2 ∗ |g|2(w)
Therefore when w ∈ [yk, yk+1] ⊆ wk + U we have
∣∣‖Sg(f)(·, w)‖22 − ‖Sg(f)(·, wj)‖2
2
∣∣ =1
2π
∣∣|f |2 ∗ |g|2(w) − |f |2 ∗ |g|2(wj)∣∣
≤ 1
2πsupu∈U
∣∣|f |2 ∗ |g|2(w) − |f |2 ∗ |g|2(w + u)∣∣
≤ 1
2π|f |2 ∗ osc(|g|2)(w)
23
This gives
∣∣∣1
2π‖f‖2
2 −∑
j
wj+1 − wj−1
2‖Sg(f)(·, wj)‖2
2
∣∣∣ ≤ 1
2π
∫ ∑
j
|f |2 ∗ osc(|g|2)(w)1[yj,yj+1](w) dw
≤ 1
2π
∫|f |2 ∗ osc(|g|2)(w) dw
≤ γ
2π‖f‖2
2
which can be rewritten to
1 − γ
2π‖f‖2
2 ≤∑
j
wj+1 − wj−1
2‖Sg(f)(·, wj)‖2
2 ≤1 + γ
2π‖f‖2
2
Now we turn to sampling in the t variable. Note that if we define g∗(x) = g(−x) theng∗(w) = g(w) and therefore
Sg(f)(t, w) =1√2π
∫f(x)g(x− t)e−ixw dx =
1√2π
(M−wf) ∗ g∗(t)
where Mwf(x) = eiwxf(x). Thus Sg(f)(t, wj) is band-limited in t and by Theorem 2.4 wesee that
(1 − δΩ
2√
2
)2
‖Sg(f)(·, wj)‖22 ≤
∑
k
tj,k+1 − tj,k−1
2|Sg(f)(tj,k, wj)|2
≤(1 +
δΩ
2√
2
)2
‖Sg(f)(·, wj)‖22
Composing with the inequality from sampling in the w variable we get
1 − γ
2π
(1 − δΩ
2√
2
)2
‖f‖22 ≤
∑
j
∑
k
(wj+1 − wj−1)(tj,k+1 − tj,k−1)
4|Sg(f)(tj,k, wj)|2
≤ 1 + γ
2π
(1 +
δΩ
2√
2
)2
‖f‖22
which finishes the proof.
Example 2.11. Notice that if the function g is not continuous then the function
Sg(g)(t, w) =1√2π
(M−wg) ∗ g∗(t)
cannot be integrable in R2. If it were then it would be integrable in t for almost all w, andthe one-dimensional Fourier transform
Ft(Sg(g)(·, w))(y) =1√2πg(y − w)g∗(y)
would be continuous in y, which is not the case.
24
We now investigate the case when g = 1[−1/2,1/2] and U = [−γ, γ] for γ < 1/4. Thenwe can choose the points wj = jγ/2 and tj,k = 4k which satisfy the requirements in The-orem 2.10 (though they are very regular). Then the local oscillation oscU(|g|2)(w) can befound explicitly
oscU(|g|2)(w) = supy∈U
|1[−1/2,1/2](w + y) − 1[−1/2,1/2](w)|
= supy∈U
|1[−y−1/2,−y+1/2](w) − 1[−1/2,1/2](w)|
= 1[−γ−1/2,γ−1/2]∪[−γ+1/2,γ+1/2](w)
and‖oscU(|g|2)‖L1 = 4γ < 1
Therefore we can apply the sampling theorem, yet the short time Fourier transform Sg(g) isnot integrable. This will be important later.
2.3.3 Group Theory
We now demonstrate how the short time Fourier transform has an underlying group structure.Let G = R × R × T be the reduced Heisenberg group with composition
(t1, w1, eiz1)(t2, w2, e
iz2) = (t1 + t2, w1 + w2, ei(z1+z2−t1w2))
and inverse given by(t, w, eiz)−1 = (−t,−w, e−itw)
Let π be the Schrodinger representation of G on L2(R) given by
π(t, w, eiz)f(x) = ei(z+wx)f(x− t)
Using the Fourier transform on L2(R) we can obtain the equivalent representation
π(t, w, eiz)f(y) = ei(z−t(y−w))f(y − w)
for f ∈ L2(R). These two representations representation are unitary. A Haar measure on Gis the measure determined by the following integral
Cc(G) 3 f 7→∫ 2π
0
∫
R
∫
R
f(t, w, eiz) dt dw dz ∈ C
For any two functions f, g in L2(R) define
Sg(f)(t, w, eiz) =1√2π
(f, π(t, w, eiz)g)L2(R) = e−iz∫
R
f(x)g(x− t)e−iwx dx
We can also view this in terms of the equivalent representation π and we then have
Sg(f)(t, w, eiz) =1√2π
(f , π(t, w, eiz)g)L2(R) =1√2πe−i(z+wt)
∫
R
f(y)g(y − w)eiyt dy
25
We now shall see that for every f, g ∈ L2(R) this function Sg(f) on the group G is squareintegrable. To do so we first carry out the following calculations (and justify their validitylater). The integral over T does not contribute in this, in fact we have
(Sg1(f1), Sg2(f2))L2 =
∫ 2π
0
∫
R
∫
R
Sg1(f1)(t, w, eiz)Sg2(f2)(t, w, eiz) dt dw dz
= 2π
∫
R
∫
R
Sg1(f1)(t, w, 1)Sg2(f2)(t, w, 1)dt dw
=
∫
R
∫
R
(∫
R
f1(y)g1(y − w)eiytdy)(∫
R
f2(y)g2(y − w)e−2πiytdy)dt dw
Let hw(y) = f1(y)g1(y − w) and kw(y) = f2(y)g2(y −w) then hw and kw are in L1(R) so the
expressions in parentheses above are F−1(hw) kw i.e.
= 2π
∫
R
∫
R
h∨w(t)kw(t) dt dw
and by the Parseval theorem we get
= 2π
∫
R
∫
R
hw(t)kw(t) dt dw
= 2π
∫
R
∫
R
f1(t)g1(t− w)f2(t)g2(t− w) dt dw
and an appliction of Fubini’s theorem gives
= 2π(f1, f2)L2(R)(g2, g1)L2(R)
If f = f1 = f2 and g = g1 = g2 then the calculations above involve only positive functionsand Fubini’s theorem is applicable. This shows that Sg(f) is in L2(G) for all f, g ∈ L2(R).Once this is established the calculation above are justified for all f1, f2, g1, g2. This actuallyalso shows that the representation π is irreducible on L2(R) (if g 6= 0 and (π(x)g, g) = 0for all x ∈ G then Sg(f) = 0 and therefore f = 0). Furthermore if we choose g such that‖g‖L2(R) = 1
2π, then we can decuce that
‖Sg(f)‖L2(G) = ‖f‖L2(R)
Inserting f2 = π(t, w, eiz)f and defining the group convolution by
F ∗G(t, w, eiz) =
∫ 2π
0
∫
R
∫
R
F (t1, w1, eiz1)G(t− t1, w − w1, e
i(w−t1w1)) dt1 dw1 dz1
we obtain the reproducing formula
Sg(f) ∗ Sg(g)(t, w, eiz) = Sg(f)(t, w, eiz) (2.3)
26
We now aim to classify the image of Sg in the case when Sg(g) ∈ L1(G). The isometryproperty ensures that the image of Sg is closed in L2(G) as we shall now see. If Sg(fn) → Fin L2(G) then Sg(fn) is Cauchy in L2(G) and fn will be Cauchy in L2(R). So fn goes tosome f ∈ L2(R) and then
‖F − Sg(f)‖L2(G) ≤ ‖F − Sg(fn)‖L2(G) + ‖Sg(fn) − Sg(f)‖L2(G)
= ‖F − Sg(fn)‖L2(G) + ‖fn − f‖L2(R)
The right hand side can be picked as small as desirable, so F = Sg(f), which shows that theimage of Sg is a closed subspace of L2(G), which we will now characterize as L2(G) ∗ Sg(g).If F is in the image of Sg(f) then F = Sg(f) = Sg(f) ∗ Sg(g) ⊆ L2(G) ∗ Sg(g). Now assumethat F ∈ L2(G) ∗ Sg(g), then for every h ∈ L2(R) the relation
(f, h) =
∫ 2π
0
∫
R
∫
R
F (t, w, eeiz
)(π(t, w, eiz)g, h) dt dw dz
is continuous in h and defines a vector f ∈ L2(R) weakly. Then Sg(f) = F ∗ Sg(g) = F byassumption and we have shown that F is in the image of Sg. Therefore we obtain that
Sg : L2(R) → L2(G) ∗ Sg(g)
is an isometric isomorphism. Further the image of Sg is a reproducing kernel Hilbert space,and the reproducing kernel is given by convolution by Sg(g). The inverse of Sg is givenweakly as
S−1g (F ) =
∫ 2π
0
∫
R
∫
R
F (t, w, eeiz
)π(t, w, eiz)g dt dw dz
Remark 2.12. (a) To conclude that the image of Sg is a reproducing kernel Hilbert sub-space of L2(G), we do not need the integrability of Sg(g). It is enough to assume thatL2(G) 3 F 7→ F ∗ Sg(g) ∈ L2(G) is continuous. This is interesting, since we have al-ready investigated sampling in the case where Sg(g) is non-integrable in Example 2.11.
(b) As a curiosity we note that we can even show that the image is a reproducing kernelHilbert space without the continuity of the map L2(G) 3 F 7→ F ∗ Sg(g) ∈ L2(G). Allthat is necessary it that the convolution F ∗ Sg(g) is well defined, which is the casesince Sg(g) ∈ L2(G). Let us go through the proof of showing that the image of Sg is areproducing kernel Hilbert space. It is enough to show the reproducing property andthat the image is a closed subspace of L2(G). Let fn be a sequence in L2(R) such thatSg(fn) → F in L2(G). We first show that F ∗ Sg(g) = F and next choose an f suchthat Sg(f) = F . Since Sg(fn) → F in L2(G) we know that there is a subsequence fnk
such that Sg(fnk)(x) → F (x) for almost all x. We then see that
|F ∗ Sg(g)(x) − F (x)| ≤ |F ∗ Sg(g)(x) − Sg(fnk)(x)| + |Sg(fnk
)(x) − F (x)|= |F ∗ Sg(g)(x) − Sg(fnk
) ∗ Sg(g)(x)| + |Sg(fnk)(x) − F (x)|
≤ ‖F − Sg(fnk)‖L2(G)‖Sg(g)‖L2(G) + |Sg(fnk
)(x) − F (x)|
27
The right hand side can be made arbitrarily small for almost every x and thus F ∗Sg(g) = F almost everywhere. The weakly defined vector
f = π(F )gdef=
∫F (x)π(x)g dx
is in L2(R), since the mapping
L2(R) 3 f 7→∫F (x)(π(x)g, f) dx ∈ C
is continuous (here we use Sg(f) ∈ L2(G)). Then Sg(f) = Sg(π(F )g) = F ∗ Sg(g) = Fand we have shown that F is in the image of Sg. Hence the image of Sg is closedand a reproducing kernel Hilbert subspace of L2(G). As noted the sampling theoremspresented in this section do not rely on F 7→ F ∗ Sg(g) being a projection.
Notice that Sg(f)(t, w, eiz) = e−izSg(f)(t, w, 1), so all information about Sg(f) can bedetermine by knowing Sg(f)(t, w, 1). Therefore we can regard Sg as a function on the ho-mogeneuous space R2 ' G/T and we obtain the short time Fourier transform or windowedFourier transform with window g
Sg(f)(t, w) =1√2π
∫
R
f(x)g(x− t)e−iwx dx
Furthermore we can reconstruct f by
f =
∫
R
∫
R
Sg(f)(t, w)π(t, w, 1)g dt dw (2.4)
When we rewrite the reproducing formula obtained with group convolution in this setting,we end up with the skew-convolution from (2.1).
This seems to have given no new information, rather we have moved from L2(R) toL2(G). What we have gained is that the functions of interest in L2(G) are contained in thereproducing kernel Hilbert space L2(G) ∗ Sg(g), so point evaluation is continuous. Also ifwe can determine Sg(f) by knowing its samples, we will be able to use equation (2.4) todetermine f .
2.4 Common Features
We now emphasize common features in the sampling theory for band-limited functions andfor the short time Fourier transform. As we have discussed, there is a group G and arepresentation π on a Hilbert space H lurking in the background of both theories. Definingthe general wavelet transform Wψ(f) = (f, π(x)ψ) for x ∈ G, then we were able to set upa correspondance between the space H and a reproducing kernel Hilbert space. The mainreasons this construction works are that
• ψ is cyclic in H
28
• L2(G) 3 F 7→ F ∗Wψ(ψ) ∈Wψ(H) ⊆ L2(G) is a continuous projection
• the reproducing formula Wψ(f) = Wψ(f) ∗Wψ(ψ) holds for all f ∈ H
We will generalize this machinery to the setting of Banach spaces, but it is our aimthat any construction has to apply to the examples of band-limited functions and the shorttime Fourier transform presented above. Work has already been done in this direction byFeichtinger and Grochenig, but it is required that the coefficients Wψ(ψ) are integrable. Thisintegrability condition fails in the band-limited case and for the short time Fourier transform(for example when the Fourier transform of the window is not continuous). Therefore wesee the need for a more general construction. Our main focus will be on the continuousdescription of the Banach spaces in question, but we point out sampling theorems and otherdiscretization methods when possible.
29
Chapter 3
Groups and Banach Spaces
We will now turn to the more general case of sampling in Banach spaces. The questionis whether it is possible to define a Banach space of distributions, a reproducing kernelBanach space of functions and an isomorphism between these two spaces. This transformarises naturally in the context of square integrable representations on Hilbert spaces, andwe describe this construction first. The transform is then extended to Banach spaces forintegrable representations with a construction due to Feichtinger and Grochenig. We describethe work of Feichtinger and Grochenig and next present the main result of this thesis, which isa generalization of their theory. Feichtinger and Grochenig derive a discretization mechanism,which relies on the integrability of the representation. We have not carried this out for thegeneral construction, but treat discretization on a per case basis in subsequent chapters.
3.1 Square Integrable Representation
As we have seen, one of the important features of the short time Fourier transform is thereproducing formula (2.3). This formula turns out to be a consequence of a more fundamentalresult, due to M. Duflo and C.C. Moore, for square integrable representations. Here weinclude the details of the proof and show how a reproducing formula arises.
Let H be a Hilbert space and π be a representation of the group G on H. The represen-tation π is said to be irreducible, if for a closed subspace V of H the inclusion π(G)V ⊆ Vimplies V = 0 or V = H. The inner product of u, v ∈ H is denoted (u, v)H (if there is noambiguity we will usualy drop the subscript H). A Radon measure µ on a locally compactgroup G is called a left invariant Haar measure if µ(xU) = µ(U) for all measurable U . Alocally compact group G admits a unique (up to multiplication by a positive constant) leftinvariant Haar measure [Mun53, Section 17] or [Wei40]. Fix such a Haar measure dx on Gand let
L2(G) =f∣∣∣∫
G
|f(x)|2dx <∞
be the space of equivalence classes of square integrable functions. By (`, L2(G)) we denotethe left-regular representation
`yf(x) = f(y−1x)
30
Definition 3.1. An irreducible unitary representation (π,H) is called square integrable, ifthere is a non-zero u ∈ H such that
∫
G
|(π(x)u, u)|2 dx <∞ (3.1)
A non-zero vector u ∈ H satisfying (3.1) is called admissible.
A self-adjoint (possibly unbounded) operator T satisfying (Tu, u) ≥ 0 for all u in itsdomain is said to be positive. The following theorem is first found in [DM76] and later theconnection to wavelet theory was made in [GMP85, GMP86].
Theorem 3.2 (Duflo-Moore). If π is square integrable then there is a positive operator Cwith dense domain D ⊆ H such that u ∈ D if and only if u is admissible. If u1, u2 ∈ D theorthogonality relation
∫
G
(π(x)u1, v1)(π(x)u2, v2)dx = 〈Cu1, Cu2〉〈v2, v1〉
Before we will prove this theorem, we will need some operator theory and the followingversion of Schur’s Lemma found in [GMP85]
Lemma 3.3. Let (π,H and (ρ,K) be two unitary representations of G and assume that πis irreducible. Suppose that T is a closed intertwining operator Tπ(x) = ρ(x)T with domainD, then D = H and T is a multiple of an isometry.
Proof. On D define the inner product
(u, v)DT= (u, v)H + (Tu, Tv)K
This is the inner product on the graph of T and since T is closed the graph is a Hilbertspace. Thus D with this inner product becomes a Hilbert space which we denote DT . Therepresentation π restricted to DT becomes unitary, but is in general not irreducible. Nowthe inclusion operator S : DT → H will be continuous, and thus there is an adjoint operatorS∗ : H → DT such that (Su, v)H = (u, S∗v)D for all v ∈ H. We then have that SS∗ : H → Hintertwines π and therefore SS∗ = λI by the usual Schur’s lemma. The mapping S is aninclusion and therefore injective, which means that since S(S∗S − λI) = (SS∗ − λI)S = 0the operator S∗S = λI on DT . Therefore
(u, u)H = (Su, Su)H = (S∗Su, u)DT= λ(u, u)DT
= λ(u, u)H + λTuTuK
for u ∈ D. This shows that λ > 0 and also that
‖Tu‖2K =
1 − λ
λ‖u‖2
H
so λ < 1 and T is a multiple of an isometry. Therefore there is an extension T of T such thatT : H → K and ‖T‖K = c‖u‖H. Each x ∈ H can be approximated by xn in D, and then
‖Txn − T x‖K = ‖T xn − T x‖K = c‖xn − x‖Hwhich shows that Txn converges to T x. Since T is closed this means that x ∈ D and Tx = T x.Therefore D = H and T is a multiple of an isometry.
31
We also need to be able to define square roots of unbounded positive operators. Thisresult relies on the existence and uniqueness of a square root of bounded positive operators,and was given an elemenary proof in [Wou66].
Lemma 3.4. Let H and K be Hilbert spaces. If T : H → K is a closed positive operatorwith domain D dense in H, then there is a unique positive operator A1/2 : H → H with thesame domain D such that
(Tu, Tv)K = (A1/2u,A1/2v)H
Proof. Let us again by DT denote the Hilbert space obtained from D with inner product
(u, v)DT= (u, v)H + (Tu, Tv)K
In the topology on DT , the operator T : DT → K is bounded and thus has an adjointT ∗ : K → DT , i.e.
(T ∗Tu, v)DT= (Tu, Tv)K
By definition of the inner product on DT we can rewrite the right hand side to obtain
(T ∗Tu, v)DT= (u, v)DT
− (u, v)H
and the linearity of the inner product gives
(u, v)H = ((1 − T ∗T )u, v)DT
Let us look at the norm of T ∗T : DT → DT . Since for ‖u‖H = ‖v‖H = 1
|(T ∗Tu, v)DT| = |(Tu, Tv)K|≤ ‖Tu‖K‖Tv‖K≤
√‖u‖2
DT− ‖u‖2
H
√‖v‖2
DT− ‖v‖2
H
< ‖u‖DT‖v‖DT
we get that‖T ∗T‖ < 1
Therefore C = 1 − T ∗T : DT → DT is invertible with bounded inverse and
(C−1u, v)H = (u, v)DT
Replacing u by T ∗Tu we get(Au, v)H = (T ∗Tu, v)DT
where A = C−1T ∗T : DT → DT . The operator A is positive and bounded, since T ∗T ispositive and bounded on DT , and therefore it has a unique positive square root A1/2 withdomain DT (see [RSN55, Theorem p. 265]) for which
(A1/2u,A1/2v)H = (Tu, Tv)DT
32
Lemma 3.5. For admissible u define the operator Tu : H → L2(G) pointwise by Tuv(x) =(v, π(x)u). Then Tu is bounded and a multiple of an isometry.
Proof. The operator Tu has domain D = v ∈ H|(v, π(·)u) ∈ L2(G).D is dense: If v ∈ D then π(y)v is also in D since
∫
G
|(π(y)v, π(x)u)|2dx =
∫
G
|(v, π(y−1x)u)|2dx =
∫
G
|(v, π(x)u)|2dx
by left invariance of the Haar measure. Therefore D is π-invariant and since 0 6= u ∈ D it isnon-zero and D must be dense.
Tu is closed: Assume that vn → v in H and that Tuvn → f in L2(G). Then
∫
G
|Tuvn(x) − f(x)|2dx→ 0
means that Tuvn converges to f in measure. Therefore there is a subsequence vnksuch that
Tuvnk(x) converges to f(x) for almost all x ∈ G. Also we know that Tuvnk
(x) → Tuv(x) forall x, since the inner product is continuous. This finally gives
|Tuv(x) − f(x)| ≤ |Tuv(x) − Tuvnk(x)| + |Tuvnk
(x) − f(x)|
The right hand side can be picked arbitarily small for almost all x ∈ G. So Tuv(x) = f(x)for almost all x, and Tuv = f in L2(G) proving that Tu is closed.
Tu intertwines π and the left regular representation `: This is seen by the unitarity of πas follows
Tu(π(y)v)(x) = (π(y)v, π(x)u) = (v, π(y−1x)u) = `yTuv(x)
D = H and Tu is a multiple of an isometry: This now follows from Schur’s lemma(Lemma 3.3).
We are now ready to prove the Duflo-Moore theorem
Proof of Theorem 3.2. We know that Tuidefined in Lemma 3.5 is a multiple of an isometry,
so ∫
G
(v1, π(x)u1)H(π(x)u2, v2)Hdx = (Tu1(v1), Tu2(v2))L2(G) = λu1λu2(v1, v2)H
where λu1 and λu2 are constants. For an non-zero v ∈ H define the operator Sv : H → L2(G)with domain D = u|
∫|(v, π(x)u)|2 dx <∞ by
Svu(x) = (v, π(x)u)
If u is admissible, then by Lemma 3.5 the vector u is in D. We have that
Sv(π(y)u)(x) = (v, π(xy)u)
and so‖Sv(π(y)u)‖2
L2(G) = ∆(y)−1‖Svu‖2L2(G)
33
Therefore D is π-invariant, non-zero and dense in H. The operator Sv is closed (same proof
as for Tu in Lemma 3.5) and thus we can construct an operator A1/2v : H → H such that
λu1λu2‖v‖2 =
∫
G
(v, π(x)u1)(π(x)u2, v) dx = (Svu2, Svu1)L2(G) = (A1/2v u2, A
1/2v u1)
A1/2v might depend on v, but we now show that this is not the case. Choose ‖v‖ = 1 then
the above readsλu1λu2 = (A1/2
v u2, A1/2v u1)
The left hand side does not depend on v so the right hand side must be independent ofthe chosen v. By uniqueness of the square root we thus get that the right hand side doesnot depend on v. This proves the existense and uniqueness of the unbounded operatorC : H → H with dense domain D with the given properties and finishes the proof.
Now we will see how square integrable representations automatically lead to a reproducingformula.
Let u be chosen such that ‖Cu‖ = 1 and define the wavelet transform
Wu(v)(x) = (v, π(x)u)
then
Wu(v) ∗Wu(u)(x) =
∫(v, π(y)u)(u, π(y−1x)u) dy
=
∫(π(y)u, v)(π(y)u, π(x)u) dy
= ‖Cu‖2(v, π(x)u)
= Wu(v)(x)
and this proves the reproducing formula
Wu(v) ∗Wu(u) = Wu(v) (3.2)
for v ∈ H. As for the Schrodinger representation the wavelet tranform Wu is an isomorphismfrom H onto a closed subspace of the space L2(G).
3.2 Feichtinger and Grochenig Theory
In [Fei83] H.G. Feichtinger constructed a family of Banach spaces, the so-called modulationspaces, using the square integrable representation which shows up in the Short Time FourierTransform. Later in [FG88, FG89a, FG89b] Feichtinger and Grochenig generalized thisconstruction to other square integrable representations, creating the coorbit space theory.Here we present the theory of Feichtinger and Grochenig without proofs to motivate our ownwork.
34
3.2.1 Quasi Banach Spaces of Functions
Let M be a measure space with measure µ. A quasi Banach space of functions is a vectorspace Y of equivalence classes of measurable functions on M for which there exists a mappingf 7→ ‖f‖Y and a constant C > 0 such that
(1) ‖f‖Y ≥ 0 and ‖f‖Y = 0 if and only if f = 0 µ-almost everywhere.
(2) ‖λf‖Y = |λ|‖f‖Y for every scalar λ
(3) ‖f1 + f2‖Y ≤ C(‖f1‖Y + ‖f2‖Y ) for all f1, f2 ∈ Y
(4) Y is complete in the topology defined by ‖ · ‖Y
A quasi Banach space of functions Y is called solid, if |f2| ≤ |f1| almost everywhere andf1 ∈ Y imply that f2 ∈ Y . The space Y is a Banach space if we can use C = 1. In examplesthroughout this thesis we will often work with σ-finite measure spaces (M, µ) and BanachSpaces of Functions Y continuously embedded in L1
loc(M, µ).A well known family of a solid quasi Banach space of functions, where µn is the Lebesgue
measure on Rn, are the spaces
Lp(Rn) =f∣∣∣‖f‖p =
(∫
R
|f(x)|p dµn(x))1/p
<∞
for 0 < p < ∞. For all these space convergence in the quasi-norm ‖ · ‖p leads to theconvergence almost everywhere for a subsequence, a property noted to be of some importancein Remark 2.12. When p ≥ 1 these spaces are continuously imbedded in L1
loc(Rn).
If (M, µ) and (N, ν) are two measure spaces and X and Y are two quasi Banach spaces offunctions on M and N respectively, we can construct a new quasi Banach space of functionsBX,Y by
BX,Y = f : M × N → C|the function (y 7→ ‖fy‖X) is in Y where fy(x) = f(x, y) for every y ∈ Y . An example of a family of function spaces constructedin this fashion is
Lp,q(Rm+n) =f∣∣∣‖f‖p,q =
(∫
Rn
( ∫
Rm
|f(x, y)|pdµm(x))q/p
dµn(y))1/q
<∞
If the measure space M is a homogeneous space on which G acts from the left and µis a measure on M invariant under the group G, then we can define left translation `y of afunction by
`yf(x) = f(y−1 · x)where y · x denotes the action of y ∈ G on x ∈ M. A quasi Banach space of functions Y onM is called left invariant if `yY ⊆ Y .
35
3.2.2 Coorbit Theory
A weight m is a continuous function m : G → R+. It is called submultiplicative if m(xy) ≤m(x)m(y) for all x, y ∈ G. For a submultiplicative weight m define the Banach space ofequivalence classes of measurable functions
Lpm(G) =f∣∣∣‖f‖Lp
m=
(∫
G
|f(x)|pm(x) dx)1/p
<∞
Let us for simplicity assume that m(x) ≥ 1 for all x ∈ G as in [Gro91, Page 5]. In thefollowing it is assumed that Y is a solid Banach Function space continuously included inL1loc(G) for which Y ∗ L1
m(G) ⊆ Y and
‖f ∗ g‖Y ≤ ‖f‖Y ‖g‖L1m
Also it is assumed that the space of analyzing vectors, defined as
Am = u ∈ H|Wu(u) ∈ L1m(G)
contains a non-zero vector. Fix a non-zero analyzing vector u ∈ Am with ‖Cu‖H = 1 anddefine
H1m = v ∈ H|Wu(v) ∈ L1
m(G)equipped with the norm ‖v‖H1
m= ‖Wu(v)‖L1
m. The space H1
m is a π-invariant Banach space,dense in H, and independent of the chosen u ∈ Am with equivalent norms (see [FG88,Lemma 4.2]). Let (H1
m)∗ be the conjugate dual of H1m with (v′, v) denoting the conjugate
dual pairing of v ∈ H1m and v′ ∈ (H1
m)∗. The conjugate dual is used instead of the usual dualto make the inner product extend in a natural way to a conjugate dual pairing. Feichtingerand Grochenig note that
Lemma 3.6. The following are continuous linear inclusions
H1m → H → (H1
m)∗
Furthermore H1m is dense in H and H is weakly dense in (H1
m)∗. Consequently the triple(H1
m,H, (H1m)∗) is a Gelfand triple.
Proof. We only have to prove continuity and denseness.The continuity of the inclusion map H1
m → H can be realized by the calculation (andthe assumption m(x) ≥ 1)
‖v‖2H = (v, v)H
=
∫
G
(π(x)u, v)H(v, π(x)u)H dx
≤ ‖u‖H‖v‖H∫
G
|(v, π(x)u)H|m(x) dx
= ‖u‖H‖v‖H‖v‖H1m
36
This also proves that H is continuously included in (H1m)∗, in particular when (H1
m)∗ isequipped with its weak topology.
From [FG88, Lemma 4.2] we already know that H1m is dense in H.
Now assume that H is not weakly dense in (H1m)∗ and let v′ be a vector in (H1
m)∗ whichis not in the weak closure of H. By the Hahn-Banach theorem ([Rud91, Theorem 3.5]) thereis a Λ in the weak dual ((H1
m)∗)′ of (H1m)∗, such that Λ(v′) = 1 and Λ(H) = 0. Since
(H1m)∗ is equipped with its weak topology, its dual ((H1
m)∗)′ can be identified with H1m which
is continuously included in H. Thus Λ can be regarded as an element in H and we get that(Λ,H)H = 0. But then Λ is zero in H and thus also zero in H1
m. This is a contradictionwith the assumption that Λ(v′) = 1.
Define the voice transform of an element v ∈ (H1m)∗ by
Wu(v′)(x) = (v′, π(x)u)
and further defineCoFGY = v ∈ (H1
m)∗|Wu(v) ∈ Y with norm ‖v‖FG = ‖Wu(v)‖Y . The space CoFGY is called a coorbit space and the followingstatements (and more) were proven in [FG89a] (see Theorem 4.2 and Proposition 4.3 in[FG89a]).
Theorem 3.7. (1) CoFGY is a π-invariant Banach space
(2) Wu : CoFGY → Y ∗Wu(u) is an isometric isomorphism
(3) If f ∈ Y ∗Wu(u) then f ∈ L∞1/m(G) and π(f)u =
∫Gf(x)π(x)u dx is well defined in
CoFGY . Furthermore CoFGY = π(Y ∗Wu(u))u.
(4) CoFGY does not depend as a set on the chosen non-zero u ∈ Am and other non-zeroanalyzing vectors give equivalent norms.
3.2.3 Discretization and Sampling
After the construction of coorbit spaces, Feichtinger and Grochenig derive a mechanism fora discrete description of these spaces. We here present the main steps in their construction.
In this context a compact neighbourhood V of e denotes a set whose interior V contains eand whose closure V is compact. In order to discretize Feichtinger and Grochenig introducedsequence spaces related to Y in the following fashion.
Definition 3.8. Let V ⊆ G be a compact neighbourhood of e. Then the sequence xiiis said to be V -separated if all the xiV are pairwise disjoint. Let U be another compactneighbourhood of e then xii is U -dense if G ⊆ ∪ixiU . A sequence xii which is bothV -separated and U -dense for some compact neighbourhoods U, V is called well-spread.
In the discretization process we will make use of the following type of partitions of unity.
37
Definition 3.9. Given a compact neighbourhood U the functions 0 ≤ ψi ≤ 1 are called aU -BUPU (bounded uniform partition of unity) if
∑i ψi = 1 and there is a well-spread family
xi such that supp(ψi) ⊆ xiU .If Ψj is an Uj-BUPU for j = 1, 2 then we write Ψ1 ≤ Ψ2 if U1 ⊆ U2. This gives a partial
order on the family of all BUPU’s and we write Ψα → 0 if Uα → e.
As proven in [Fei81, Theorem 2] we can always construct an U -BUPU for any givenwell-spread U -dense set xi.
We now introduce the sequence spaces related to Y . The definition makes use of acompact neighbourhood U , but the resulting spaces have equivalent norms for different Uand different well-spread sequences xi (see [FG89a, Lemma 3.5]). What we present here issimpler than what Feichtinger and Grochenig do. They work with finite unions of well-spreadsequences which are U -dense, while we only deal with one well-spread sequence.
Definition 3.10. Assume the points xii∈I are well-spread and U is a compact neighbour-hood of e. For any sequence of numbers Λ = λii∈I , let the function fΛ =
∑i∈I |λi|1xiU be
the pointwise limit of partial sums. We then define the sequence space Yd to be
Yd = Λ = λii∈I |fΛ ∈ Y
with norm ‖λi‖Yd= ‖
∑i |λi|1xiU‖Y .
Remark 3.11. (a) We are not sure whether this is exactly the definition used by Feichtingerand Grochenig or if they define the limit of the sum to be in norm. In the latter case thisposes a problem, since then `∞ 6= (L∞(G))d. To see this let λi = 1; then the partial sums∑n
i=1 |λi|1xiU do not converge in L∞. However when we use a pointwise definition of the sumthen for Y = Lpm(G) the sequence space is Yd = `pm′ where m′(i) = m(xi) for all 1 ≤ p ≤ ∞.
(b) The solidity of Y ensures that the indicator functions of compact sets are in Y andthus the finite sequences are contained in Yd. But are the finite sequences dense in Yd?The answer depends on Y in the following sense: Assume that λi is in Yd and definef(x) =
∑I λi1xiU(x) to be a pointwise limit of partial sums (this pointwise limit exists
since the xiU ’s have finite overlap). This function f is in Y by solidity. If the compactlysupported continuous functions are dense in Y , then this convergence will also take placein the norm of Y as noted in [Rau05][Lemma 4.3.1 (c)]. To see this choose a compactlysupported continuous function g close to f in Y . Then the support of g can be covered byfinitely many xiU ’s, supp(g) ⊆ ∪i∈I′xiU . The partial sum fI′(x) =
∑i∈I′ λi1xiU agrees with
f on a set larger than the support of g and thus |f − fI′| ≤ |f − g| everywhere. The solidityof Y then ensures that ‖f − fI′‖Y ≤ ‖f − g‖Y , which shows that the partial sums are dense.This implies that if the bounded compactly supported functions are dense in Y , then thefinite sequences are dense in Yd, and further we can interpret the sum
∑I as a norm limit
of partial sums.
We next introduce three discretization operators first found in [FG88] and [Gro91]. Theseoperators can be regarded as a sort of Riemann sums for the functions in Y ∗Wu(u).
Definition 3.12. Given a well-spread sequence xi, which is U -dense and a U -BUPU ψifor which supp(ψi) ⊆ xiU we define the operators (though we do not know yet if they arewell-defined)
38
(1) TU1 : Y ∗Wu(u) → Y ∗Wu(u) by
TU1 f =∑
i
f(xi)ψi ∗Wu(u)
(2) TU2 : Y ∗Wu(u) → Y ∗Wu(u) by (with ci =∫ψi)
TU2 f =∑
i
cif(xi)`xiWu(u)
(3) TU3 : Y ∗Wu(u) → Y ∗Wu(u) by
TU3 f =∑
i
(∫f(x)ψi(x) dx
)`xiWu(u)
If the compactly supported continuous functions are dense in Y the sums above are inter-preted with convergence in norm, otherwise the convergence is pointwise.
For a function G we define the U -oscillation G#U by
G#U (x) = sup
y∈U|G(yx) −G(x)|
With this we are able to formulate the following version of a collection of theorems from[Gro91] stating when the operators are well defined and invertible
Theorem 3.13. Assuming we can choose u ∈ H1w such that the U -oscillations Wu(u)
#U are
in L1w then the operators TU1 , T
U2 , T
U3 : Y ∗Wu(u) → Y ∗Wu(u) are well-defined and converge
to the identity on Y ∗Wu(u) in operator norm as U → e.
When these operators are invertible they provide Banach frames and atomic decomposi-tions as described in the following
Definition 3.14 (Banach frames). Let B be a Banach space with dual B∗ then a familyeii∈I ⊆ B∗ is a Banach frame for B if there is a Banach sequence space Bd such that
(i) ‖f‖B and ‖〈ei, f〉‖Bdare equivalent
(ii) and there is a bounded operator S : Bd → B such that S(〈ei, f〉) = f .
Theorem 3.15 (Banach frames). Assume that Wu(u)#U ∈ L1
w and U is small enough thatTU1 and TU2 are invertible, then the vectors ei = π(xi)u form a frame for CoFGY withsequence space Yd, and we have two reconstruction operators as listed below.
39
(TU1 ) Any v ∈ CoFGY can be reconstructed by
v = W−1u (TU1 )−1
[∑
i
Wu(v)(xi)ψi ∗Wu(u)]
If the bounded compactly supported functions are dense in Y then
v =∑
i
Wu(v)(xi)ei
where ei = W−1u (TU1 )−1[ψi ∗Wu(u)] is a “dual frame”.
(TU2 ) Any v ∈ CoFGY can be reconstructed by
v = W−1u (TU2 )−1
[∑
i
ciWu(v)(xi)ψi ∗Wu(u)]
If the bounded compactly supported functions are dense in Y then
v =∑
i
Wu(v)(xi)ei
where ei = W−1u (TU2 )−1[ψi ∗Wu(u)] is a “dual frame”.
A discretization framework which facilitates series expansions can be defined as
Definition 3.16 (Atomic decompositions). Let B be a Banach space with dual B∗ then thefamilies fii∈I ⊆ B and gii∈I ⊆ B∗ form an atomic decomposition for B, if there is aBanach sequence space Bd such that
(i) the mapping B 3 f 7→ 〈gi, f〉 ∈ Bd is continuous,
(ii) the mapping Bd 3 λi 7→ ∑i λifi ∈ B is continuous
(iii) and f can be reconstructed by f =∑
i∈I〈gi, f〉fk
The convergence of the sums above is not necessarily in norm, but in any suitable topology.
Theorem 3.17 (Atomic decompositions). Assume that Wu(u)#U ∈ L1
w and U is small enoughthat TU2 and TU3 are invertible.
(TU2 ) Let fi = π(xi)u and 〈gi, v〉 = [(TU2 )−1Wu(v)](xi), then fi and gi form an atomicdecomposition for CoFGY with sequence space Yd.
(TU3 ) Let fi = π(xi)u and 〈gi, v〉 =∫ψi(x)((T
U3 )−1Wu(v))(x) dx, then fi and gi form
an atomic decomposition for CoFGY with sequence space Yd.
In both cases v ∈ CoFGY can be reconstructed by v =∑
i∈I〈gi, v〉fi, where the convergenceis in norm if the bounded compactly supported functions are dense in Y and weak* in (H1
m)∗
otherwise.
40
3.2.4 Examples
We here present examples which are covered by the coorbit theory by Feichtinger andGrochenig. These spaces were shown to be coorbits in the papers [FG88, FG89a, Gro91].The modulation spaces were introduced in [Fei83], and the book [Gro01] is a good source ofinformation about modulation spaces.
Modulation Spaces
The short time Fourier transform from section 2.3 has a generalization to n-dimensions
Sg(f)(t, w) =1√2π
n (f, π(t, w, 1)g)
whereπ(t, w, eiz)g(x) = ei(z+w·x)g(x− t)
is a representation on L2(Rn) of the group G = Rn × Rn × T with composition
(t1, w1, eiz1)(t2, w2, e
iz2) = (t1 + t2, w1 + w2, ei(z1+z2−t1·w2))
Choose g in the space S(Rn) of rapidly decreasing smooth functions, then Sg(f) can beextended to tempered distributions f ∈ S ′(Rn) by
Sg(f)(t, w) =1√2π
n 〈f, π(t, w, 1)g〉
Let m be a weight on R2n satisfying m(t, w) ≤ C(1 +√|t|2 + |w|2)s for some s ≥ 0. For
1 ≤ p, q <∞ define the norm
‖f‖Mp,qm
= ‖Sg(f)‖Lp,qm
=(∫ (∫
|Sg(f)(t, w)m(t, w)|p dt)q/p
dw)1/q
on the modulation spaces Mp,qm
Mp,qm = f ∈ S ′(Rn)|‖f‖Mp,q
m<∞
These spaces are Banach spaces and they are coorbits for the reduced Heisenberg group G.
Besov Spaces
Let φ ∈ S(Rn) be a radial function satisfying
• 0 ≤ φ ≤ 1
• φ(w) > 0 for |w| ∈ [23, 4
3]
• supp(φ) ⊆ w||w| ∈ [13, 5
3]
•∑
j φ(2jw) = 1 for all w 6= 0
41
Define φj ∈ S(Rn) by φj(w) = φ(2−jw), then for a tempered distributions f the convolutionf ∗ φj is a well-defined function. For 1 ≤ p, q <∞ define the norm
‖f‖Bp,qs
=(∑
j
2jsq‖f ∗ φj‖qp)1/q
Then the homogeneous Besov space Bp,qs is the space of tempered distributions
Bp,qs = f ∈ S(Rn)|‖f‖Bp,q
s<∞
Note that if f(x) = xml (m an integer) then
f ∗ φj(y) =
∫(yl − xl)
mφj(x) dx
=
m∑
k=0
(m
k
)(−y)n−k
∫xkl φj(x) dx
=m∑
k=0
(m
k
)(−y)n−kik
( ∂
∂wl
)kφj(0)
= 0
This means that ‖f‖Bp,qs
= 0 for all polynomials f , and thus the space Bp,qs consists equiv-
alence classes modulus polonomials. With this identification the Besov spaces are Banachspaces (see [Pee76, Tri88a, FJW91]).
The Besov spaces have been shown to be coorbits for the group G = (R+O(n))oRn withthe representation
π(A, b)f(x) =1√
det(A)f(A−1(x− b))
The rotational component vanishes when φ is radial, and as shown in [Gro91, p. 11] (withmore details in [Rau05, Section 4.7.2])
Bp,qs = CoFGL
p,qs+n/2−n/q(G) = f ∈ S(Rn)|Wφ(f) ∈ Lp,qs+n/2−n/q(G)
with equivalent norms, when
Lp,qs (G) =f∣∣∣( ∫
R
(∫
Rn
|f(a, b)|p db)q/p
a−sqda
an+1
)1/q
<∞
As noted we only need to work with the subgroup H = R+ oRn for which the representationis not irreducible, yet φ as chosen will be a cyclic vector for the representation.
3.3 Coorbit Spaces for Dual Pairs
In section 3.1 we showed that a square integrable representation leads to a reproducingformula. Yet, recently it has been proven by Zimmermann in [Zim05] that reproducing
42
formulas can also occur in the case of non-unitary representations. We have further metnon-irreducible representation in the context of band-limited functions and in the case ofBesov space in section 3.2.4. Therefore we will give a formulation of coorbit theory whichdoes neither require the representations to be irreducible nor unitary. In the Feichtinger andGrochenig theory the integrability of the representation is used to define the intermediatespace H1
m. This is done in order to get a large enough pool of distributions (H1m)∗ to be able
to define the coorbit space. In applications the Banach space (H1m)∗ is often replaced by
a Frechet space invariant under the representation π (see the examples of modulation andBesov spaces in section 3.2.4). We will investigate how the choice of Frechet space influencesthe construction of coorbits. We present our suggestion for a generalized coorbit theory inthis section.
Let S be a Frechet space and let S∗ be the space of continuous conjugate linear functionalson S equipped with the weak topology. We assume that S is continuously imbedded andweakly dense in S∗. The conjugate dual pairing of elements v ∈ S and v′ ∈ S∗ will bedenoted by 〈v′, v〉.
Let G be a locally compact group with a fixed left Haar measure dx, and assume that(π, S) is a representation of G. Also assume that the representation is continous,i.e. g 7→π(g)v is continuous for all v ∈ S. As usual define the contragradient representation (π∗, S∗)by
〈π∗(x)v′, v〉 = 〈v′, π(x−1)v〉.Then π∗ is a continuous representation of G on S∗. For a fixed vector u ∈ S define the linearmap Wu : S∗ → C(G) by
Wu(v′)(x) = 〈v′, π(x)u〉.
The map Wu is called the voice transform or the wavelet transform.
Assumption 3.18. Let Y be a left invariant Banach Space of Functions on G, and assumethat there is a non-zero cyclic vector u ∈ S satisfying the following properties
(R1) the reproducing formula Wu(v) ∗Wu(u) = Wu(v) is true for all v ∈ S
(R2) the space Y is stable under convolution with Wu(u) and f 7→ f ∗Wu(u) is continuous
(R3) if f = f ∗Wu(u) ∈ Y then the mapping S 3 v 7→∫f(x)〈π∗(x)u, v〉 dx ∈ C is in S∗
(R4) the mapping S∗ 3 v′ 7→∫〈v′, π(x)u〉〈π∗(x)u, u〉 dx ∈ C is weakly continuous
A vector u satisfying Assumption 3.18 is called an analyzing vector. Note that (R4)implies that there is an element v ∈ S such that
〈v′, v〉 =
∫〈v′, π(x)u〉〈π∗(x)u, u〉 dx
for all v′ ∈ S∗. This ensures that the vector v ∈ S can be weakly defined by
v = π(Wu(u)∨)u =
∫
G
Wu(u)∨(x)π(x)u dx
where we have used the notation f∨(x) = f(x−1).
43
Theorem 3.19. Assume that Y and u satisfy Assumption 3.18 and define the coorbit space
CouSY = v′ ∈ S∗|Wu(v′) ∈ Y (3.3)
equipped with the norm ‖v′‖ = ‖Wu(v′)‖Y . Then the following properties hold
(1) Wu(v) ∗Wu(u) = Wu(v) for v ∈ CouSY .
(2) The space CouSY is a π∗-invariant Banach space.
(3) Wu : CouSY → Y intertwines π∗ and left translation
(4) The convolution operator f 7→ f ∗Wu is a bounded projection from Y to the closedsubspace Wu(CouSY ) = Y ∗Wu(u).
(5) CouSY = π∗(f ∗Wu(u))u|f ∈ Y .
(6) Wu : CouSY → Y ∗Wu(u) is an isometric isomorphism
Proof. (1) The space S is weakly dense in S∗, so pick a net vα in S for which 〈vα, v〉 → 〈v′, v〉for all v ∈ S. By assumption (R1) the reproducing formula Wu(vα) ∗Wu(u) = Wu(vα) istrue for each vα. The continuity requirement (R4) gives that
v′ 7→∫〈π∗(y−1)v′, π(x)u〉〈π∗(x)u, u〉 dx
=
∫〈v′, π(x)u〉〈u, π(x−1y)u〉 dx
= Wu(v′) ∗Wu(u)(y)
is weakly continuous. Therefore Wu(vα) ∗Wu(u)(y) → Wu(v′) ∗Wu(u)(y) for every y ∈ G.
By assumption vu(vα)(y) → vu(v′)(y) for all y ∈ G, and we conclude that
Wu(v′)(y) = Wu(v
′) ∗Wu(u)(y) for all y ∈ G.
This reproducing formula is valid for all v′ ∈ S∗ and therefore also for v′ ∈ CouSY ⊆ S∗.(2,3) We now check that ‖v′‖ = ‖Wu(v
′)‖Y is indeed a norm. The only non-obviousquestion is if ‖v′‖ = 0 gives v′ = 0. If ‖v′‖ = 0 then ‖Wu(v
′)‖Y = 0 and so 〈v′, π(x)u〉 = 0for almost all x. The function x 7→ 〈v′, π(x)u〉 is continuous which shows that it is identically0 for all x. But u is cyclic in S, so this implies that 〈v′, v〉 = 0 for all v ∈ S, and thus v′ = 0in S∗. This also proves the injectivity of Wu.
Assume that vn is a Cauchy sequence in CouSY , then Wu(vn) is a Cauchy sequence in Yso Wu(vn) converges to a function f ∈ Y . Then
‖f ∗Wu(u) − f‖Y ≤ ‖f ∗Wu(u) −Wu(vn) ∗Wu(u)‖Y + ‖Wu(vn) ∗Wu(u) − f‖Y= ‖(f −Wu(vn)) ∗Wu(u)‖Y + ‖Wu(vn) − f‖Y
The second term goes to 0, since f is the limit in Y of Wu(vn). The mapping Y 3 g 7→g ∗ Wu(u) ∈ Y is continuous,so it follows that the first term also tends to zero and thusf = f ∗Wu(u). Then by assumption (R3) an element v′ ∈ S∗ can be defined by
〈v′, v〉 =
∫f(x)〈π∗(x)u, v〉 dx
44
and it follows that
Wu(v′)(y) = 〈v′, π(y)u〉
=
∫f(x)〈π∗(x)u, π(y)u〉 dx
=
∫f(x)〈u, π(x−1y)u〉 dx
= f ∗Wu(u)(y)
= f(y)
which shows that v′ ∈ CouSY .Assume that v′ is in CouSY , then the voice transform of π∗(y)v′ is
Wu(π∗(y)v′)(x) = 〈π∗(y)v′, π(x)u〉 = 〈v′, π(y−1x)u〉 = `yWu(v
′)(x)
Since Wu(v′) ∈ Y and the space Y is assumed to be left invariant, it follows that Wu(π
∗(y)v′)is in Y . This shows that CouSY is π-invariant, and also that Wu intertwines π and lefttranslation, thereby proving both (2) and (3).
(4) The subspace Y ∗Wu(u) is closed since if fn ∗Wu(u) → f in Y then
f = limn→∞
(fn ∗Wu(u))
= limn→∞
(fn ∗Wu(u) ∗Wu(u))
= ( limn→∞
fn ∗Wu(u)) ∗Wu(u)
= f ∗Wu(u)
The second last equality is valid, because Y 3 f 7→ f ∗Wu(u) ∈ Y is continuous.(6) The injectivity of Wu follows from the fact that ‖v′‖ = ‖Wu(v
′)‖Y is a norm. Wenow show that Wu(CouSY ) = Y ∗Wu(u). If v′ ∈ CouSY then Wu(v
′) ∈ Y and also Wu(v′) =
Wu(v′) ∗Wu(u) ∈ Y ∗Wu(u). If on the other hand f ∈ Y ∗Wu(u) then f = f ∗Wu(u) and
assumption (R3) again tells us that there is a v′ ∈ S∗ defined by
〈v′, v〉 =
∫f(x)〈π∗(x)u, v〉 dx
for v ∈ S. Direct calculation shows that
Wu(v′) = f ∗Wu(u) = f ∈ Y
such that v′ ∈ CouSY . Therefore Wu : CouSY → Y ∗ Wu(u) is surjective. That Wu is anisometry follows directly from the definition of the norm.
(5) Above we have shown that for f ∈ Y there is a v′ ∈ CouSY such that v′ = π(f∗Wu(u))u.If on the other hand v′ ∈ CouSY then let f = Wu(v
′) = f ∗Wu(u) ∈ Y ∗Wu(u). Then by(R3) π∗(f)u defines an element in S∗ and
〈π∗(f)u, π(y)u〉 =
∫f(x)〈π∗(x)u, π(y)u〉 dx
= f ∗Wu(u)(y)
= f(y)
= 〈v′, π(y)u〉
45
This shows that π∗(f)u and v′ agree for all π(y)u, and since u is cyclic in S, it follows thatπ∗(f ∗Wu(u))u = π∗(f)u and v′ are the same element in S∗.
Remark 3.20. (a) The proof above can readily be generalized to work also for quasiBanach spaces, and in this case CouSY is a quasi Banach space.
(b) Assume the properties in Assumption 3.18 can be verified for a solid Banach Function
space Y , and that the (possibly non-solid) subspace Y is continuously included in Y .
Then Y has an associated coorbit space if the continuity (R2) can be proven. Also
CouSY is continuously included in CouSY . In section 3.4.2 we will see how this can be
done when Y is a Sobolev space on a Lie group.
(c) As hinted in Remark 2.12 (b) it is a question whether we need the continuity assumption(R2) from Assumption 3.18. It seems to be possible to replace it with the assumptionthat if fn → f in the space Y then there is a subsequence fnk
which converges pointwiseto f almost everywhere.
(d) Theorem 4.2(i) in [FG89a] states that CoFGY is continuously included in (H1m)∗, and
Theorem 4.5.13(d) in [Rau05] states further that H1m is continuously included in CoFGY .
It is an open problem whether similar statements are true for S,CouSY and S∗.
The following theorem tells us which analyzing vectors will give the same coorbit space.
Theorem 3.21. If u1 and u2 both satisfy Assumption 3.18 and for i, j ∈ 1, 2 the followingproperties can be verified
• there are non-zero constants ci,j such that Wui(v) ∗Wuj
(ui) = ci,jWuj(v) for all v ∈ S
• Y 3 f 7→ f ∗Wui(uj) ∈ Y is continuous
• S∗ 3 v′ 7→∫〈v′, π(x)ui〉〈π∗(x)ui, uj〉 dx ∈ C is weakly continuous
then Cou1S Y = Cou2
S Y with equivalent norms.
Proof. Assume that u1 and u2 are two analyzing vectors, i.e. they satisfy the propertiesAssumption 3.18. We claim first that
Wu1(v) ∗Wu2(u1) = c1,2Wu2(v)
for all v ∈ S∗. With v ∈ S this is true by the assumption. The space S is weakly dense inS∗ and therefore the identity Wu1(v) ∗Wu2(u1) = c1,2Wu2(v) is true for all v ∈ S∗. This isverified by applying the third continuity condition to the integral
Wu1(v) ∗Wu2(u1)(y) =
∫〈π∗(y−1)v, π(x)u1〉 〈π∗(x)u1, u2〉 dx
If Wu1(v) ∈ Y then Wu1(v) ∗Wu2(u1) = c1,2Wu2(v) ∈ Y , since Y ∗Wu2(u1) is assumed tobe a subset of Y . Symmetry then gives us that Cou1
S Y = Cou2S Y .
It remains to show that the norms ‖v‖1 = ‖Wu1(v)‖Y and ‖v‖2 = ‖Wu2(v)‖Y are equiv-alent norms on Cou1
S Y = Cou2S Y . We have assumed that the mappings f 7→ f ∗ Wu2(u1)
46
and f 7→ f ∗ Wu2(u1) are continuous. This means that ‖f ∗ Wu1(u2)‖Y ≤ A1‖f‖Y and‖f ∗Wu2(u1)‖Y ≤ A2‖f‖Y . But then
c2,1‖v‖1 = c2,1‖Wu1(v)‖Y = ‖Wu2(v) ∗Wu1(u2)‖Y ≤ A1‖Wu2(v)‖Y = A1‖v‖2
Similarly c1,2‖v‖2 ≤ A2‖v‖1 which shows the norms are equivalent.
In the following we will describe how the choice of the Frechet space S affects the coorbitspace. We will show that there is great freedom when choosing S.
Theorem 3.22. Let S and T be Frechet spaces which are weakly dense in their conjugateduals S∗ and T ∗ respectively. Let π and π denote representations ofG on S and T respectively.Assume there is a vector u ∈ S and u ∈ T such that the requirements in Assumption 3.18are satisfied by both (u, S) and (u, T ). Also assume that the conjugate dual pairings ofS∗ × S and T ∗ × T satisfy 〈u, π(x)u〉S = 〈u, π(x)u〉T for all x ∈ G. Then CouSY and CouTYare isometrically isomorphic. The isomorphism is given by WuW
−1u .
Proof. Let Wu(v′)(x) = 〈v′, π(x)u〉S for v′ ∈ CoSuY and Wu(v
′)(x) = 〈v′, π(x)u〉T for v′ ∈CoTuY . Since it is assumed that Wu(π(x)u) = Wu(π(x)u) for all x ∈ G the spaces CoSuYand CoTuY are both isometrically isomorphic to the space Y ∗ Wu(u) = Y ∗ Wu(u). Theisomorphism between CouSY and CouTY is exactly W−1
u Wu : CoSuY → CoTuY .
Let π be a unitary irreducible representation of G on H. Assume that the Frechet spacesS and T are π-invariant and that (S,H, S∗) and (T,H, T ∗) are Gelfand triples with thecommon Hilbert space H. Then S ∩ T is π-invariant and if we can pick a non-zero vectoru ∈ S ∩ T , such that u is analyzing for both S and T , then
〈u, π(x)u〉S = (u, π(x)u)H = 〈u, π(x)u〉T
and we are in the situation of the previous theorem. We summarize the statement as
Corollary 3.23. Assume that (S,H, S∗) and (T,H, T ∗) are Gelfand tripples and assumethere is an analyzing vector u ∈ S∩T such that both (u, S) and (u, T ) satisfy Assumption 3.18for some Banach space Y , then CouSY and CouTY are isometrically isomorphic.
If the Frechet space S is a dense subspace of the Frechet space T , and S is continuouslyincluded in T , then we can regard the space T ∗ as a subspace of S∗. With this identificationthe two coorbit spaces will be equal. We state the following
Theorem 3.24. Let (π,H) be a unitary irreducible representation of G, and let (S,H, S∗)and (T,H, T ∗) be Gelfand triples for which (π, S) and (π, T ) are representations of G. As-sume that i : S → T is a continuous linear inclusion and that there is u ∈ S such that both(u, S) and (i(u), T ) satisfy Assumption 3.18. Then the map i∗ restricted to Co
i(u)T Y is an
isometric isomorphism between Coi(u)T Y and CouSY .
Proof. Since the vector i(u) is assumed cyclic in T , we see that i(S) is dense in T , andtherefore i∗ : T ∗ → S∗ is injective. This allows us to view T ∗ as a subspace of S∗.
47
Let Wu(v′)(x) = 〈v′, π(x)u〉S and Wi(u)(v
′) = 〈v′, π(x)i(u)〉T . If v′ ∈ CouSY then
Wu(v′) ∗Wi(u)(i(u))(x) =
∫Wu(v
′)(y)(i(u), π(y−1x)i(u))H dy (3.4)
=
∫Wu(v
′)(y)(u, π(y−1x)u)H dy
= Wu(v′) ∗Wu(u)
= Wu(v′)
so (R3) tells us there is an element v′ ∈ T ∗ such that for v ∈ T
〈v′, v〉T =
∫Wu(v
′)(x)〈π∗(x)i(u), v〉T dx
It is true that i∗(v′) = v′ in S∗, since
〈i∗(v′), π(x)u〉S = 〈v′, π(x)i(u)〉T = 〈v′, π(x)u〉Sfor each x ∈ G, and u is cyclic. The found element v′ is unique. This shows that CouSY ⊆i∗(Co
i(u)T Y ).
If on the other hand v′ ∈ Coi(u)T Y then
Wu(i∗(v′))(x) = 〈i∗(v′), π(x)u〉S = 〈v′, π(x)i(u)〉T = Wi(u)(v
′)(x) ∈ Y
which shows that i∗(v′) ∈ CouSY . This shows that i∗(Coi(u)T Y ) ⊆ CouSY .
That the mapping i∗ is an isometry when restricted to Coi(u)T Y follows directly from the
calculations in (3.4).
Remark 3.25. If (π, S) is a representation of G and u is a cyclic vector for which it is truethat 〈π∗(x)u, u〉 = 〈u, π(x)u〉 for all x ∈ G and both (R1) and (R4) are satisfied, then 〈v, w〉is an inner product on S. The completion H of S with respect to the norm ‖v‖H =
√〈v, v〉
is a Hilbert space. The representation π will then extend to a unitary representation π onH, but we will not be able to conclude that π is irreducible. Therefore the construction ofcoorbit spaces also works for non-irreducible representations, as long as we choose a cyclicvector in the Frechet space S.
Note also that a reproducing formula has been constructed from a non-unitary represen-tation in [Zim05], thus allowing for construction of coorbit spaces in this new setting.
The following theorem is a slight generalizaton of [FG89a, Theorem 4.9], which in theoryenables us to apply it to more general coorbit spaces, than the ones treated in [FG89a]. Theproof follows that of [FG89a, Theorem 4.9], but we include it here for completeness.
Theorem 3.26. Let Y ∗ be the conjugate dual space of Y and assume it is also a Banachspace of functions. Assume that u ∈ S is a vector satisfying Assumption 3.18 for both Yand Y ∗. If the conjugate dual pairing on Y ∗ × Y satisfies
〈f ∗Wu(u), g〉Y ∗×Y = 〈f, g ∗Wu(u)〉Y ∗×Y (3.5)
then (CouSY )∗ = CouS(Y∗). If Y is reflexive so is CouSY .
48
If the conjugate dual pairing of Y and Y ∗ is the extension of an integral then property(3.5) is true.
Proof. Define a linear map T : CouS(Y∗) → (CouSY )∗ by
〈Tw′, v′〉Y ∗×Y = 〈Wu(w′),Wu(v
′)〉(CouSY )∗×Cou
SY
for w′ ∈ CouS(Y∗) and v′ ∈ CouSY . The map T is a well defined, since Wu is a topological
isomorphism onto its image.If T (w′) = 0 for some w′ ∈ CouS(Y
∗) then for any f ∈ Y we have
〈Wu(w′), f〉Y ∗×Y = 〈Wu(w
′) ∗Wu(u), f〉Y ∗×Y= 〈Wu(w
′), f ∗Wu(u)〉Y ∗×Y= 〈Tw′,W−1
u (f ∗Wu(u))〉(CouSY )∗×Cou
SY
= 0
since f∗Wu(u) ∈Wu(CouSY ). SoWu(w′) = 0 in Y ∗, and by the injectivity ofWu : CouS(Y
∗) →Y ∗ ∗Wu(u) we conclude that w′ = 0. This shows that T is injective.
Let w′ ∈ (CouSY )∗ and define f ∈ Y ∗ by
〈f , g〉Y ∗×Y = 〈w′,W−1u (g ∗Wu(u))〉(Cou
SY )∗×CouS(Y )
for all g ∈ Y . Notice thatf ∗Wu(u) = f
which can be seen by the calculation
〈f ∗Wu(u), g〉Y ∗×Y = 〈f , g ∗Wu(u)〉Y ∗×Y= 〈w′,W−1
u (g ∗Wu(u) ∗Wu(u))〉(CouSY )∗×Cou
SY
= 〈w′,W−1u (g ∗Wu(u))〉(Cou
SY )∗×CouSY
= 〈f , g〉Y ∗×Y
Thus there is a w′ ∈ CouS(Y∗) such that
f = Wu(w′)
Finally for all v′ ∈ CouSY the calculation
〈Tw′, v′〉(CouSY )∗×Cou
SY= 〈Wu(w
′),Wu(v′)〉Y ∗×Y
= 〈f ,Wu(v′)〉Y ∗×Y
= 〈w′,W−1u (Wu(v
′) ∗Wu(u))〉Y ∗×Y= 〈w′,W−1
u (Wu(v′))〉Y ∗×Y
= 〈w′, v′〉(CouSY )∗×Cou
SY
shows that T (w′) = w′ and proves that T is surjective.
49
3.4 Lie Groups and Smooth Representations
In this section we point out why smooth representations are easy to work with. Furthersmooth representations provide us with an family of examples for the generalized coorbittheory. We construct these spaces, but do not give a discrete description.
3.4.1 Smooth Square Integrable Reprensentations
Let (π,H) be a square integrable representation of a Lie group G. This means that π isunitary and irreducible and that there is a non-zero u such that
∫
G
|(u, π(x)u)H|2 dx <∞
The material covered here can be found in for example [War72].Let g be the Lie algebra of G and choose a basis X1, . . . , Xn for g. A function f : G 7→ H
is called differentiable, if the directional derivative defined by
Xf(x) =d
dt
∣∣∣t=0f(exp(−tX)x)
exists and is continuous for all X ∈ g. A function is called smooth if directional derivatives ofall orders are continuous, and the space of such functions is denoted C∞(G,H). The smoothfunctions will be equipped with the topology given by uniform convergence of directionalderivatives on compact sets. This topology makes C∞(G,H) into a Frechet space, and isinduced by the seminorms
‖f‖m,K = supx∈K
‖Xi1Xi2 . . .Ximf(x)‖H | ik = 1, . . . , n
A vector v is called smooth for the representation π if the mapping x 7→ π(x)v is of classC∞(G,H). The space of smooth vectors is denoted H∞
π and is a Frechet space with thetopology inherited from C∞(G,H). Denote by H−∞
π the conjugate dual of H∞π . An X ∈ g
induces the differential operator
π(X)v =d
dt
∣∣∣t=0π(exp(tX))v
defined for v ∈ H∞π . There is a special collection of smooth vectors which are invariant
under π(X). This space consists of vectors (called Garding vectors) of the form π(f)v =∫f(x)π(x)v dx for f ∈ C∞
c (G) and v ∈ H and is called the Garding space. These vectorssatisfy
π(X)π(f)v = π(Xf)v
and are thus invariant under π(X).Since π is assumed square integrable the reproducing formula (3.2) holds in particular
for v ∈ H∞π , and the following result is used to extend the reproducing formula to v ∈ H−∞
π .
50
Proposition 3.27. If u ∈ H∞π is in the domain of the operator C from Theorem 3.2, then
the map
H−∞π 3 v′ 7→
∫(v′, π(x)u)(π(x)u, u) dx ∈ C
is continuous in the weak topology, and thus (R4) is satisfied.
The next two results state, that if there is a non-zero u ∈ H satisfying (R1) and (R2),then there is a smooth non-zero vector which satisfies the same conditions.
Proposition 3.28. Let f ∈ C∞c (G) then if u ∈ H satisfies (R1), then so will a constant
multiple of π(f)u.
Proof. Let C be the operator from Theorem 3.2 with domain D(C). Assume that u ∈ D(C)and f ∈ C∞
c (G). Then
Wπ(f)u(π(f)u)(z) =
∫ ∫f(x)f(y)(π(y)u, π(zx)u)H dx dy
= f ∗Wu(u) ∗ (f∨)(z)
where f∨(x) = f(x−1). Since f, f∨ ∈ Cc(G) ⊆ L1(G) and Wu(u) ∈ L2(G), the functionf ∗Wu(u) ∗ f∨ is in L2(G). This shows that π(f)u is in the domain of C which finishes theproof.
Proposition 3.29. Assume that Y is a solid Banach Function space, and that Y 3 F 7→F ∗ g ∈ Y is continuous for each g ∈ C∞
c (G). Further assume that there is a u ∈ H suchthat Y 3 F 7→ F ∗ |Wu(u)| ∈ Y is continuous. Then any Garding vector of the form π(f)usatisfies (R2).
Proof. The solidity of Y and the fact that Y ∗ |Vu(u)| ⊆ Y justifies changing order ofintegration, so by direct calculation it can be shown that
F ∗ Vπ(f)u(π(f)u)(x) =
∫ ∫ ∫F (z)f(y)f(w)(π(y)u, π(z−1xw)u)H dy dw dz
= F ∗ f ∗ Vu(u) ∗ f∨(x)
where f∨(x) = f(x−1). The continuities in the assumptions then ensure that the mappingY 3 F 7→ F ∗ Vπ(f)u(π(f)u) ∈ Y is continuous.
Remark 3.30. Similar results can be shown for the space Hωπ of analytic vectors, and its
dual H−ωπ , by approximating the Garding vector π(f)u by an analytic vector cf. [War72, p.
278 ff.].
With these results in place the Gelfand triples (H∞π ,H,H−∞
π ) and (Hωπ ,H,H−ω
π ) will benatural choices for the construction of some coorbit spaces.
51
3.4.2 Coorbits for Sobolev Spaces
As mentioned in Remark 3.20(b) it is also possible to construct coorbits for non-solid Banachspaces. Let Y be a solid Banach space, (π, S) a representation and u ∈ S a vector such
that Assumption 3.18 is satisfied. Then we can construct coorbits for any Banach space Ycontinuously included in Y , whenever it is possible to prove continuity of the mapping
Y 3 f 7→ f ∗Wu(u) ∈ Y
Here we will present the particular case where Y is a Sobolev space on a Lie group.Following [Tri87, Tri88b] we define the Sobolev spaces Wp
m(G) to be the space of functionsfor which
‖f‖Wpm
=
m∑
k=0
∑
1≤li≤n‖Xl1 · · ·Xlkf‖p <∞
This is a Banach space, yet it is not solid for m > 0, since the indicator functions are not inWp
m(G).
Proposition 3.31. Let (π, S) be a representation of G, and u ∈ S a vector such thatAssumption 3.18 is satisfied with Y = Lp(G). Assume that S ⊆ H∞
π and π(X)S ⊆ S forall X ∈ g. If for each v ∈ S there is a compact interval I around 0 such that the functiongv(y) = sups∈I |Wu(π(exp(−sX)v)(y)| is in (Lp(G))∗, then CouSWp
m(G) exists.
Proof. Assume that f,Xf ∈ Lp(G) for X ∈ g then
X(f ∗Wu(u))(x) =d
dt
∣∣∣t=0
∫f(y)Wu(u)(y
−1 exp(−tX)x) dy (3.6)
The integration and differentiation can be swapped if for each x there is an interval I suchthat
sups∈I
∣∣∣d
dt
∣∣∣t=sf(y)Wu(u)(y
−1 exp(−tX)x)∣∣∣
is integrable. Now look at
sups∈I
∣∣∣d
dt
∣∣∣t=sWu(u)(y
−1 exp(−tX)x)∣∣∣ = sup
s∈I
∣∣∣d
dt
∣∣∣t=s
(π(y)u, π(exp(−tX)x)u)∣∣∣
= sups∈I
∣∣∣(π(y)u, π(exp(−sX)π(X)π(x)u)∣∣∣
= sups∈I
|Wu(π(exp(−sX)π(X)π(x)u)(y)|
By assumption this is less than gπ(X)π(x)u(y) if we choose I small enough. Since we have alsoassumed that gv(y) is in (Lp(G))∗ we are allowed to swap differentiation and integration and
52
obtain
X(f ∗Wu(u))(x) =d
dt
∣∣∣t=0
∫f(y)Wu(u)(y
−1 exp(−tX)x) dy
=d
dt
∣∣∣t=0
∫f(exp−tXy)Wu(u)(y
−1x) dy
=
∫(Xf)(y)Wu(u)(y
−1x) dy
= (Xf) ∗Wu(u)(x)
Repeating this argument we see that
Wpm(G) 3 f 7→ f ∗Wu(u) ∈ Wp
m(G)
is continuous. The rest of the properties for constructing coorbits are satisfied since Wpm(G) ⊆
Lp(G) for which the coorbit space is assumed to exist.
53
Chapter 4
The Special Linear Group andBergman Spaces
4.1 SL2(R), SU(1, 1) and Discrete Series Representations
The group SL2(R) is the group of 2 × 2-matrices with determinant 1
SL2(R) =
(a bc d
) ∣∣∣ad− bc = 1
SL2(R) has the Iwasawa decomposition KAN where
K =
kθ =
(cos θ sin θ− sin θ cos θ
)∣∣∣θ ∈ R
' SO(2),
A =
at =
(t 00 t−1
) ∣∣∣t 6= 0
,
N =
nr =
(1 r0 1
) ∣∣∣r ∈ R
and the map(kθ, at, nr) 3 K ×A×N 7→ kθatnr ∈ SL2(R)
is a diffeomorphism. Conjugation by the matrix
C =1√2
(1 −i−i 1
)
yields the group SU(1, 1)
SU(1, 1) =
(α ββ α
) ∣∣∣|α|2 − |β|2 = 1
Written in detail the Cayley map Φ : SL2(R) → SU(1, 1) looks like
Φ
(a bc d
)=
1
2
(1 −i−i 1
)(a bc d
) (1 ii 1
)=
1
2
(a+ d+ i(b− c) b+ c + i(a− d)b+ c− i(a− d) a+ d− i(b− c)
)
54
The Iwasawa decomposition of SU(1, 1) is then KΦAΦNΦ where
KΦ =
(eiθ 00 e−iθ
)∣∣∣θ ∈ R
' SU(1),
AΦ =
(cosh t i sinh t
−i sinh t cosh t
) ∣∣∣t 6= 0
,
NΦ =
(1 + ir rr 1 − ir
) ∣∣∣r ∈ R
The group SU(1, 1) acts transitively on the unit disc D = z = u+ iv ∈ C||z| < 1 by(α ββ α
).z =
αz + β
βz + α
and the SU(1, 1) invariant measure on D is dz(1−|z|2)2 where dz is the area measure. Therefore
the representations defined for integers n ≥ 2
πn
(α ββ α
)f(z) =
1
(−βz + α)nf( αz − β
−βz + α
)
are unitary on the corresponding Hilbert spaces
L2n(D) =
f∣∣∣∫
D
|f(z)|2(1 − |z|2)n−2 dz <∞
These representations are not irreducible, but if we restrict to the holomorphic functionsHn = A2
n(D) in L2n(D) the representations (πn,Hn) are irreducible (see for example [Lan85]
p. 184 ff). Composing these representations with the Cayley transform we define the repre-sentations (πn,Hn)
πn
(a bc d
)f(z) = πn
(Φ
(a bc d
) )f(z)
For the remainder of this chapter, we will only need to work with the subgroup G = ANof SL2(R). This group can be represented as
G = (a, b)|a > 0, b ∈ R
with composition(a, b)(a1, b1) = (aa1, ab1 + a−1
1 b)
We restrict the representation πn to G and write πn(a, b) for πn
(a b0 a−1
)We now argue
that the function ψ = 1D is cyclic in Hn. Any element of SL2(R) can be written as nratkθ,and therefore
πn(nratkθ)ψ(z) = πn(nrat)πn(kθ)ψ(z) = e−inθπn(nrat)ψ(z)
Since ψ is SL2(R)-cyclic (by irreducibility of πn) any vector v ∈ Hn has the sum representa-tion
v =∞∑
m=1
cme−inθmπn(nrmatm)ψ =
∞∑
m=1
dmπn(nrmatm)ψ
55
(for suitable nrm and atm) showing that ψ is also G-cyclic. We will often write an element(a, b) ∈ G as an element of SU(1, 1)
(α ββ α
)= Φ
(a b0 a−1
)=
1
2
(a+ a−1 + ib b+ i(a− a−1)b− i(a− a−1) a + a−1 − ib
)
without specifying the dependence of α and β on a and b.
4.2 Coorbits for Discrete Series
Let ψ = 1D and define the wavelet coefficients
W nψ (f)(a, b) = (f, πn(a, b)ψ)n
then
W nψ (ψ)(a, b) =
∫
D
ψ(z)πn(a, b)ψ(z)(1 − |z|2)n−2 dz
=1
π
∫ 1
0
∫ 2π
0
(α− βre−iθ)−n dθ (1 − r2)n−2r dr
= 21
αn
∫ 1
0
(1 − r2)n−2r dr
=2n
n− 1(a+ a−1 − ib)−n
Lemma 4.1. The wavelet coefficients W nψ (ψ) for n ≥ 2 satisfy
∫|W n
ψ (ψ)(a, b)|pasda dba2
<∞
for p ≥ 1 if and only if 2 − np < s < np.
Proof. In the calculation
2−n(n− 1)
∫|W n
ψ (ψ)(a, b)|pas da dba2
=
∫as−2
((a+ a−1)2 + b2)pn/2da db
=
∫(a+ a−1)as−2
((a+ a−1)2 + (a + a−1)b2)pn/2da db
=
∫ ∞
0
as−2(a+ a−1)1−np da
∫1
(1 + b2)pn/2db
the second integral is finite if pn/2 > 1/2 (which is always the case). Now look at the firstintegral and split it up
∫ ∞
0
as−2(a+ a−1)1−np da =
∫ 1
0
as+np−3
1 + a2da
︸ ︷︷ ︸finite iff s+ np > 2
+
∫ ∞
1
as−np−1
1 + a−2da
︸ ︷︷ ︸finite iff s− np < 0
56
This shows that the representations πn are square integrable for all n ≥ 2 and integrablefor n ≥ 3. That π2 is not integrable turns out to not matter for the construction of coorbitspaces. We now construct coorbit spaces for the representations πn related to the spaces
Lp(G) =F
∣∣∣∫
|F (a, b)|pda dba2
<∞
It is convenient to first present a few lemmas
Lemma 4.2. The mappings f 7→ f ∗W nψ (ψ) and f 7→ f ∗ |W n
ψ (ψ)| are continuous Lp(G) →Lp(G) for np > 2.
Proof. In the following denote by Fn the wavelet coefficient belongning to πn, i.e. Fn(a, b) =(ψ, πn(a, b)ψ). In the calculations below we make some assumptions in order for the estimatesto be true. At the end of the proof we collect these assumptions and find conditions for themto be simultaneously true.
Let p > 1 and assume that f ∈ Lp(G). Let q such that 1/p + 1/q = 1 and further let sbe such that the following calculations hold (we will investigate this later)
∣∣∣∫ ∫
f(a, b)Fn((a, b)−1(a1, b1))
da db
a2
∣∣∣p
≤(∫ ∫
|f(a, b)|a−s|Fn((a, b)−1(a1, b1))|1/p+1/qasda db
a2
)p
≤(∫ ∫
|f(a, b)|pa−sp|Fn((a, b)−1(a1, b1))|da db
a2
)
×(∫ ∫
at/pasq|Fn((a, b)−1(a1, b1))|da db
a2
)p/q
By the unitarity of πn it is true that |Fn((a, b))| = |Fn((a, b)−1)| so the second integralbecomes
∫ ∫asq|Fn((a1, b1)
−1(a, b))| da dba2
=
∫ ∫(aa1)
sq|Fn((a, b))|da db
a2
≤ Casq1
provided that Fn(a, b)asq ∈ L1(G). Thus we get
∣∣∣∫ ∫
f(a, b)Fn((a, b)−1(a1, b1))
da db
a2
∣∣∣p
≤ C
∫ ∫|f(a, b)|pa−sp|F2((a, b)
−1(a1, b1))|da db
a2asp1 .
57
Now we can calculate an estimate for the norm using Fubini’s theorem
‖f ∗ Fn‖Lp ≤ C
∫ ∫ ∫ ∫|f(a, b)|pa−sp|Fn((a, b)−1(a1, b1))|
da db
a2asp1
da1db1a2
1
= C
∫ ∫|f(a, b)|pa−sp
∫ ∫|Fn((a, b)−1(a1, b1))|asp1
da1 db1a2
1
da db
a2
= C
∫ ∫|f(a, b)|pa−sp
∫ ∫|Fn((a1, b1))|(aa1)
sp da1 db1a2
1
da db
a2
= C
∫ ∫|f(a, b)|pa−sp
∫ ∫|Fn((a1, b1))|(aa1)
sp da1 db1a2
1
da db
a2
≤ C
∫ ∫|f(a, b)|pa−spasp da db
a2
= C‖f‖pLp
where we in the second last inequality have assumed that Fn(a, b)asp ∈ L1(G).
We now gather the assumptions made during the calculations and determine when theyare all true. To sum up the map f 7→ f ∗Fn is continuous if Fn(a, b)a
sq ∈ L1 and Fn(a, b)asp ∈
L1. This is the case if both 2 − n < sq < n and 2 − n < sp < n, which can be rewritten tothe requirement that we can find an s such that
s ∈[(2 − n)(p− 1)
p,n(p− 1)
p
]∩
[2 − n
p,n
p
]
which is possible whenever 2 < pn.For p = 1 we see that L1 ∗ Fn ⊆ L1 if Fn ∈ L1 (Young’s inequality), and by lemma 4.1
this is true if n > 2. So the result also holds for p = 1.
Remark 4.3. Note that if we wish to use Young’s inequality, i.e. Lp∗L1 ⊆ Lp, then we haveto require Fn ∈ L1. By the lemma 4.1 we then require n > 2. This is more restrictive thanthe result above, and we can therefore describe a wider range of spaces than using L1-theory.
In the sequel we will need the following characterization of the smooth vectors for thediscrete series representations. These have been characterized in [OØ88] and more generallyin [CF04].
Lemma 4.4. The smooth vectors H∞n for πn are the power series
∑∞k=0 akz
k for which therefor any m holds that
∞∑
k=0
|ak|2(n− 1)!k!
(n+ k − 1)!(n(n− 2) + 2k2)2m <∞
The conjugate duals (A2n)
−∞ of these spaces are the formal power series∑∞
k=0 bkzk for which
there is an m such that
∞∑
k=1
|bk|2(n− 1)!k!
(n + k − 1)!(n(n− 2) + 2k2)−2m <∞
58
Lemma 4.5. Given f ∈ Lp the mapping
H∞n 3 φ 7→
∫ ∫f(a, b)W n
ψ (φ)(a, b)da db
a2
is continuous for 1 ≤ p <∞.
Proof. Let φ be a smooth vector with expansion∑∞
k=0 akzk. The Taylor series for the vector
πn
(α ββ α
)1(z) is
πn
(α ββ α
)1(z) = α−n
∞∑
k=0
(β/α)k(−1)k(n+ k − 1)!
(n− 1)!k!zk
Therefore, since (zk, zk)n = k!(n− 2)!/(k + n− 1)!,
W nψ (φ)(a, b) =
1
(n− 1)αn
∞∑
k=0
(−1)k(β/α)kak = 2Fn(a, b)
∞∑
k=0
(−1)k(β/α)kak
and it can estimated by
|W nψ (φ)(a, b)| ≤ 2|Fn(a, b)|
∞∑
k=0
|ak|
since |α| > |β|. For any m we obtain that
∞∑
k=0
|ak| ≤ |a0| +( ∞∑
k=1
(n+ k − 1)!
k!(n− 1)!(n(n− 2) + 2k2)−2m
)1/2
×( ∞∑
k=1
|ak|2(n− 1)!k!
(n+ k − 1)!(n(n− 2) + 2k2)2m
)1/2
and by picking m large enough the first sum converges. Therefore |W nψ (φ)| ≤ Cφ|Fn| and
the constant Cφ depends continuously on φ.Thus we only need to require that the integral
∫ ∫f(a, b)|Fn(a, b)|
da db
a2
is finite. Let 1 < p <∞ and 1 < q <∞ be such that 1/p+ 1/q = 1. The following estimateholds if f ∈ Lp and nq > 2
∫ ∫|f(a, b)||Fn(a, b)|
da db
a2≤ ‖f‖Lp‖Fn‖Lq
‖Fn‖Lq is finite when nq > 2, which is the case for all q > 1.For p = 1 we use that fact that the representation is unitary and so ‖Fn‖L∞ ≤ ‖ψ‖2, so
the same inequality as above holds with a slight modification.
59
Remark 4.6. Notice that we have not used f = f ∗W nψ (ψ) as listed in Assumption 3.18.
Theorem 4.7. The spaces CoψH∞n Lp are well-defined when pn > 2.
Proof. (R1) The reproducing formula can be verified, since the representations πn are squareintegrable. Let u be a normalization of ψ in order to make convolution with W n
u (u) anidempotent.
(R4) By Proposition 3.27 the condition (R4) is automatically satisfied when u is a con-stant multiple of ψ ∈ H∞
n .By the previous lemmas the rest of the conditions in Assumption 3.18 are satisfied for
some u ∈ H∞n (constant multiple of ψ(z) = 1) with the Banach space Y = Lp(G) when
np > 2. Therefore the space
CouH∞n Lp(G) = v′ ∈ H−∞
n |Wu(v′) ∈ Lp(G)
is a Banach space.
In the next section we will prove that these spaces defined in Theorem 4.7 are in factBergman spaces. This was mentioned in [FG88, Section 7], but not many details were given.
4.3 Bergman Spaces
For 1 ≤ p <∞ and σ > 1 the space Lpσ(D) are the measurable functions for which the norm
‖f‖Lpσ(D) =
(∫
D
|f(z)|p(1 − |z|2)σ−2 dz)1/p
The Bergman spaces are the holomorphic functions in Lpσ(D), i.e.
Apσ(D) =f ∈ O(D)
∣∣∣‖f‖pApσ(D)
=
∫
D
|f(z)|p(1 − |z|2)σ−2 dz <∞
We collect a few facts about Bergman spaces corresponding to Corollary 1.5 and Theorem1.10 in [HKZ00]
Theorem 4.8. (a) For f ∈ Apσ(D) this identity holds
f(z) = (σ − 1)
∫
D
f(w)(1 − |w|2)σ−2
(1 − zw)σdw
(b) For τ > 1 define
Pτf(z) = (τ − 1)
∫
D
f(w)(1 − |w|2)τ−2
(1 − zw)τdw
then Pτ : Lpσ(D) → Apσ(D) is a bounded projection onto Apσ(D) if and only if σ − 1 <(τ − 1)p.
60
Before proving the next theorem we need to be able to rewrite integrals over the groupG as integrals over the disc D. The left-invariant measure on G = AN is given by
∫
G
f(x) dx =
∫
R×R+
f(a, b)da db
a2
Also there is a 1 − 1 correspondance φ between G and D given by
φ(a, b) =a2 + b2 − 1
(1 + a)2 + b2︸ ︷︷ ︸x
+i−2b
(1 + a)2 + b2︸ ︷︷ ︸y
with inverse
φ−1(x+ iy) =( 1 − x2 − y2
(1 − x)2 + y2
︸ ︷︷ ︸a
,−2y
(1 − x)2 + y2
︸ ︷︷ ︸b
)
The Jacobian matrix for φ−1 is
Jφ−1(x+ iy) =2
[(1 − x)2 + y2]2
((1 − x)2 − y2 −2y(1 − x)−2y(1 − x) y2 − (1 − x)2
)
with determinant
|Jφ−1(x+ iy)| =4
[(1 − x)2 + y2]2
Thus we can pass from integrals on G to integrals on D and back by∫
G
f(a, b)da db
a2= 4
∫
D
f φ−1(z)dz
(1 − |z|2)2
We are also in need of a lemma
Lemma 4.9. If f ∈ Apnp/2 for np > 2 then f ∈ H−∞n .
Proof. We need to estimate the coefficients bk where f(z) =∑∞
k=0 bkzk. For this let us first
estimate f (k)(0). Denote by σ = np/2 and τ = dnp/2e then by Theorem 4.8 we know that
f(z) = (τ − 1)
∫
D
f(w)(1 − |w|2)τ−2
(1 − zw)τdw
Differentiate under the integral sign k times (which is allowed when for example |z| ≤ 1/2)
f (k)(z) = (τ − 1)τ(τ + 1) . . . (τ + k − 1)
∫
D
f(w)(1 − |w|2)τ−2
(1 − zw)τ+kwk dw
We then see that for z = 0 we have
|f (k)(0)| ≤ (τ − 1)τ(τ + 1) . . . (τ + k − 1)
∫
D
|f(w)|(1− |w|2)τ−2 dw
≤ (τ − 1)τ(τ + 1) . . . (τ + k − 1)‖f‖Apσ‖1‖Aq
σ
61
Therefore the coefficients bk can be estimated by
|bk| =|f (k)(0)|
k!≤ (τ − 1)τ(τ + 1) . . . (τ + k − 1)
k!≤ Ckτ
Lastly we estimate the sum
∞∑
k=1
|bk|2(n− 1)!k!
(n+ k − 1)!(n(n− 2) + 2k2)−2m ≤ C
∞∑
k=1
k2τ (n− 1)!k!
(n + k − 1)!(n(n− 2) + 2k2)−2m
≤ 4−mC
∞∑
k=1
(k2)τ−2m
and we notice that if we choose m = τ ≥ 1 this series converges. Thus f ∈ H−∞n .
Theorem 4.10. The spaces Apnp/2(D) correspond to the coorbits CouH∞n Lp(G) from Theo-
rem 4.7.
Proof. Assume that f ∈ Apnp/2(D). We already know that f ∈ H∞n ,so we can find the wavelet
coefficient of f
W nψ (f)(a, b) =
∫
D
f(z)πn(a, b)ψ(z)(1 − |z|2)n−2 dz
=
∫
D
f(z)1
(−βz + α)n(1 − |z|2)n−2 dz
=1
αn
∫
D
f(z)1
(1 − βαz)n
(1 − |z|2)n−2 dz
=n− 1
αnf(βα
)
In the last step we applied Theorem 4.8 (b) for σ = np/2 and τ = n for which σ−1 < (τ−1)p.Now β/α can be rewritten as
β
α=
2(ab)
(1 + a2)2 + (ab)2+ i
(a2)2 + (ab)2 − 1
(1 + a2)2 + (ab)2= iφ(a2, ab)
Therefore
W nψ (f)(a, b) =
n− 1
(a+ a−1 − ib)nf(iφ(a2, ab))
Then taking Lp(G)-norm of W nψ (f) and using the changing to an integral over the disc we
62
get
1
(n− 1)p
∫
G
|W nψ (f)(a, b)|pda db
a2=
∫
G
1
[(a+ a−1)2 + b2]np/2|f(iφ(a2, ab))|pda db
a2
=1
2
∫
G
1
[(√a+
√a−1
)2 + b2]np/2|f(iφ(a,
√ab))|da db
a√a
=1
2
∫
G
1
[(√a+
√a−1
)2 + (√a−1b)2]np/2
|f(iφ(a, b))|pda dba2
=1
2
∫
G
[ a
(1 + a)2 + b2
]np/2|f(iφ(a, b))|pda db
a2
= 2
∫
D
(1 − |z|2)np/2|f(iz)|p dz
(1 − |z|2)2
= ‖f‖pAp
np/2
We now show that an element of the coorbit space is in the Bergman space. The smoothvectors H∞
n is weakly dense in the dual H−∞n . Since any f ∈ H∞
n is in A2n(D) we know that
f satisfies Theorem 4.8 and
W nψ (f)(a, b) =
1
αnf(βα
)
By the weak denseness the same equality thus holds for the wavelet coefficient for f ∈H−∞n . Therefore the calculations above are valid and if f ∈ CouH∞n L
p(G) then f is also inApnp/2(D).
4.4 Discretization
The key to finding atomic decompositions will be the following result
Lemma 4.11. For each ε > 0 there is a neighbourhood U of the identity such that
∣∣∣Fn((a, b)(x, y))
Fn(x, y)− 1
∣∣∣ < ε
for (a, b) ∈ U .
Proof. Consider the expression
∣∣∣Fn((a, b)(a1, b1))
Fn(a1, b1)− 1
∣∣∣ =∣∣∣(ψ, πn(aa1, ab1 + a−1
1 b)ψ)
(ψ, πn(a1, b1)ψ)− 1
∣∣∣
Since ψ is in the Bergman space A2n(D), we have
(ψ, πn(a, b)ψ)n =1
αn1=
1
(a + a−1 − ib)n
63
Therefore, if we write α2 = aa1 + (aa1)−1 − i(ab1 + a−1
1 b), we have
∣∣∣Fn((a, b)(a1, b1))
Fn(a1, b1)− 1
∣∣∣ =∣∣∣( α1
α2
)n− 1
∣∣∣
=∣∣∣( α1
α2
)n−1
+( α1
α2
)n−2
+ · · · +( α1
α2
)+ 1
∣∣∣∣∣∣α1
α2
− 1∣∣∣
≤∣∣∣α1
α2
− 1∣∣∣n−1∑
k=0
∣∣∣α1
α2
∣∣∣k
Choose γ > 1 and δ = minγ− 1, 1/γ, and let (a, b) satisfy 1γ< a < γ and −δ < b < δ. We
first estimate
∣∣∣α1
α2
∣∣∣2
=(a1 + a−1
1 )2 + b21(aa1 + (aa1)−1)2 + (ab1 + a−1
1 b)2
≤ (a1 + a−11 )2
γ−2(a1 + a−11 )2
+b21
(aa1 + (aa1)−1)2 + (ab1 + a−11 b)2
≤ γ2 +b21
(aa1 + (aa1)−1)2 + (ab1 + a−11 b)2
If |b1| ≤ 2|aa1|−1|b| ≤ 2γδ/a1 we can use δ < 1/γ to get
b21(aa1 + (aa1)−1)2 + (ab1 + a−1
1 b)2≤ 4γ2δ2a1
−2
(aa1 + (aa1)−1)2≤ 4γ4δ2 ≤ 4γ2
If on the other hand |b1| ≥ 2|aa1|−1|b| then |ab1 + a−11 b| ≥ ||ab1| − |a−1
1 b|| ≥ 12|ab1| and so
b21(aa1 + (aa1)−1)2 + (ab1 + a−1
1 b)2≤ b21
(ab1 + a−11 b)2
≤ 4b21|ab1|2
≤ 4γ2
We thus get the bound ∣∣∣α1
α2
∣∣∣2
≤ 5γ2 (4.1)
We now turn to estimate∣∣∣α1
α2− 1
∣∣∣ =∣∣∣α1 − α2
α2
∣∣∣
=∣∣∣a1 + a−1
1 − ib1 − [aa1 + (aa1)−1 − i(ab1 + a−1
1 b)]
aa1 + (aa1)−1 − i(ab1 + a−11 b)
∣∣∣2
=∣∣∣(1 − a)a1 + (1 − a−1)a−1
1 + i((a− 1)b1 + a−11 b)
aa1 + (aa1)−1 + i(ab1 + a−11 b)
∣∣∣
≤ |1 − a|∣∣∣
a1
aa1 + (aa1)−1
∣∣∣ + |1 − a−1|∣∣∣
a−11
aa1 + (aa1)−1
∣∣∣
+ |a− 1|∣∣∣
b1
aa1 + (aa1)−1 + i(ab1 + a−11 b)
∣∣∣ + |b|∣∣∣
a−11
aa1 + (aa1)−1
∣∣∣
64
we have previously shown that∣∣∣ b1aa1+(aa1)−1+i(ab1+a−1
1 b)
∣∣∣ < 2γ which we use to get
≤ γ|1 − a| + γ|1 − a−1| + 2|a− 1|γ + γ|b|
Also 1 − 1γ≤ γ − 1 and δ < γ − 1, so we get
∣∣∣α1
α2− 1
∣∣∣ ≤ 5γ(γ − 1) (4.2)
The inequalities (4.1) and (4.2) together show that
∣∣∣Fn((a, b)(a1, b1))
Fn(a1, b1)− 1
∣∣∣ ≤∣∣∣α1
α2
− 1∣∣∣n−1∑
k=0
∣∣∣α1
α2
∣∣∣k
→ 0 as γ → 1
From this result follows easily
Corollary 4.12. There exist a neighbourhood U of the identity and constants C1, C2 > 0such that
C1|Fn(x, y)| ≤ |Fn((a, b)(x, y))| ≤ C2|Fn(x, y)|for all (x, y) ∈ G with (a, b) ∈ U . These constants can be chosen arbitrarily close to 1, bychoosing U small enough.
Proposition 4.13. Let V ⊆ U be compact neighbourhoods of the identitiy. Assume thatthe points xi are V -separated and U -dense and that U satisfies Corollary 4.12. Let ψibe a partition of unity for which supp(ψi) ⊆ xiU . Then the following is true
1. The mapping `p 3 (λi) 7→∑
i λi`xiFn ∈ Lp(G) ∗ Fn is continuous
2. The mapping Lp(G) ∗ Fn 3 f 7→ (f(xi))i∈I ∈ `p(I) is continuous
3. The mapping Lp(G) ∗ Fn 3 f 7→ (∫Gf(x)ψi(x)dx)i∈I ∈ `p(I) is continuous
As in [Gro91] sums are understood as limits of the net of partial sums over finite subsetswith convergence in Lp(G).
Proof. First note that the norms on `p and Lp(G) are related in the following sense
‖(λi)‖`p =1
|V |∥∥∥
∑
i
λi1xiV
∥∥∥Lp
The convolution in Lp(G) with |Fn| is continuous (see Lemma 4.2) and we will denote thenorm of this convolution by Dp. Further assume that we have chosen U and constants C1, C2
satisfying Lemma 4.11 and Corollary 4.12. (1) If (λi) ∈ `p then the function
f =∑
i
|λi|1xiV
65
is in Lp(G) and ‖f‖Lp(G) = |V | ‖(λi)‖`p. Convolution with |Fn| is continuous so
f ∗ |F2| =∑
i
|λi|1xiV ∗ |F2|
is in Lp(G). Now let us show that the function 1xiV ∗ |Fn| is bigger than some constant times`xi
|Fn|. ∫1xiV (z)|Fn(z−1y)|dz =
∫
V
|Fn(z−1x−1i y)|dz ≥ C1|V ||Fn(x−1
i y)|
This shows that∣∣∣∑
i
λi`xiFn(y)
∣∣∣ ≤∑
i
|λi||Fn(x−1i y)|
≤ 1
C1|V |∑
i
|λi|1xiV ∗ |Fn|(y)
=1
C1|V |f ∗ |Fn|.
Since f ∗ |Fn| ∈ Lp(G) the sum∑
i λi`xiFn is in Lp(G) with norm
∥∥∥∑
i
λi`xiFn
∥∥∥Lp
≤ 1
C1|V |‖f ∗ |Fn| ‖Lp
≤ Dp
C1|V |‖f‖Lp
=Dp
C1
‖(λi)‖`p.
This shows the desired continuity. The sum∑
i λi`xiFn is to be understood as a limit in
Lp(G) and since convolution with Fn is continuous we get from the reproducing formula that∑i λi`xi
Fn ∈ Lp(G) ∗ Fn.(2) We need to show that f(xi) is in `p, but this is the same as showing that g =∑i |f(xi)|1xiV is in Lp(G). But f ∈ Lp(G) ∗ Fn so
∑
i
|f(xi)|1xiV (y) ≤∑
i
|f | ∗ |Fn|(xi)1xiV (y) =
∫|f(z)|
∑
i
|Fn(z−1xi)|1xiV (y)dz
For each y at most one i adds to this sum, namely the i for which xi ∈ yV −1. Therefore
∑
i
|Fn(z−1xi)|1xiV (y) ≤ supv∈V
|Fn(z−1yv−1)| ≤ C2|Fn(z−1y)|
by Corollary 4.12. We then get
∑
i
|f(xi)|1xiV (y) ≤ C2
∫|f(z)||Fn(z−1y)|dz = C2|f | ∗ |Fn|(y)
66
and finally
‖f(xi)‖`p ≤ C2Dp
|V | ‖f‖Lp.
(3) We have to show that the function
∑
i
(∫f(x)ψi(x)dx
)1xiV ∈ Lp(G)
We get that
∣∣∣∑
i
(∫f(x)ψi(x)dx
)1xiV (y)
∣∣∣ ≤∫
|f(x)|∑
i∈Iψi(x)1xiV (y)dx
and since ∑
i
ψi(x)1xiV (y) ≤∑
i
1xiU(x)1xiV (y) ≤ 1U−1V (x−1y)
we obtain∣∣∣∑
i
(∫f(x)ψi(x)dx
)1xiV (y)
∣∣∣ ≤∫
|f(x)|1U−1V (x−1y)dx = |f | ∗ 1U−1V (y)
which is in Lp(G) with norm continuously dependent on f , i.e.
∥∥∥∑
i
(∫f(x)ψi(x)dx
)1xiV
∥∥∥ ≤ C‖f‖Lp
for some C > 0.
Proposition 4.14. We can choose a compact neighbourhood U , U -dense points xi and apartition ψi of unity with supp(ψi) ⊆ xiU such that the operators defined below are invertiblewith continuous inverses
1. define T1 : Lp(G) ∗ Fn → Lp(G) ∗ Fn by
T1f =∑
i
f(xi)ψi ∗ Fn
2. define T2 : Lp(G) ∗ Fn → Lp(G) ∗ Fn by (with ci =∫ψi)
T2f =∑
i
cif(xi)`xiFn
3. define T3 : Lp(G) ∗ Fn → Lp(G) ∗ Fn by
T3f =∑
i
(∫f(x)ψi(x) dx
)`xiFn
67
Proof. For each neighbourhood of the identity U we can pick U -dense points xi such thatxi are V -separated for some compact neighbourhood of the identity V satisfying V 2 ⊆ U(see [Rau05, Thm 4.2.2]). Thus we can pick U in order to satisfy the inequality in Lemma 4.11for any ε.
Denote by Dp the Lp operator norm of convolution by Fn.(1) Let f ∈ Lp(G) ∗ Fn and let us look at the difference
f(x) −∑
i
f(xi)ψi(x) =∑
i
(f(x) − f(xi))ψi
For x ∈ supp(ψi) ⊆ xiU we get
|f(x) − f(xi)| ≤∫
|f(z)||Fn(z−1x) − Fn(z−1xi)|dz
=
∫|f(z)|
∣∣∣Fn(z
−1xi)
Fn(z−1x)− 1
∣∣∣|Fn(z−1x)|dz ≤ ε
∫|f(z)||Fn(z−1x)|dz
= ε|f | ∗ |Fn|(x)This means that
|f(x) −∑
i
f(xi)ψi(x)| ≤ ε∑
i
|f | ∗ |Fn|(x)ψi(x) = ε|f | ∗ |Fn|(x)
This function is in Lp(G) and so∥∥∥f −
∑
i
f(xi)ψi
∥∥∥Lp
≤ εDp‖f‖Lp.
Convoluting this expression by Fn we get∥∥∥f −
∑
i
f(xi)ψi ∗ Fn∥∥∥Lp
≤ εD2p‖f‖Lp
So picking U such that ε < D−2p we obtain an operator T1 such that ‖I − T1‖ < 1 as an
operator on Lp(G) ∗ Fn. Therefore T1 is invertible.(2) We will show that T2 is invertible using its difference from the operator T1.
|T2f(x) − T1f(x)| =∣∣∣∑
i
f(xi)(ψi ∗ Fn(x) − ciFn(x−1i x))
∣∣∣
≤∑
i
|f(xi)||ψi ∗ Fn(x) − ciFn(x−1i x)|
Look at
|ψi ∗ Fn(x) − ciFn(x−1i x)| =
∣∣∣∫ψi(z)(Fn(z
−1x) − Fn(x−1i x)dz
∣∣∣
≤∫ψi(z)|Fn(z−1x) − Fn(x
−1i x)|dz
≤ ε
∫ψi(z)|Fn(z−1x)|dz
= εψi ∗ |Fn|(x)
68
Then we have|T2f(x) − T1f(x)| ≤ ε
∑
i
|f(xi)|ψi ∗ |Fn|(x)
This is a function in Lp(G) and the norm is
‖T2f − T1f‖Lp ≤ εDp
∥∥∥∑
i
|f(xi)|ψi∥∥∥Lp
≤ ε2D2p‖f‖Lp
This means that ‖I −T2‖ ≤ ‖I−T1‖+ ‖T1−T2‖ ≤ εDp(1+ εDp) and if we pick U such thatthis norm is less than 1 we get that T2 is invertible.
(3) We use the same trick as above, to obtain
|f(x) − T3f(x)| =∣∣∣∫f(y)Fn(y
−1x) dy −∑
i
∫f(y)ψi(y) dyFn(x
−1i x)
∣∣∣
=∣∣∣∫ ∑
i
f(y)ψi(y)Fn(y−1x) dy −
∑
i
∫f(y)ψi(y) dyFn(x
−1i x)
∣∣∣
=∣∣∣∫ ∑
i
(f(y)ψi(y))(Fn(y−1x) − Fn(x
−1i x)) dy
∣∣∣
≤∫ ∑
i
|f(y)|ψi(y)|Fn(y−1x) − Fn(x−1i x)| dy
≤∫ ∑
i
|f(y)|ψi(y)ε|Fn(y−1x)| dy
= ε
∫|f(y)||Fn(y−1x)| dy
= ε|f | ∗ |Fn|(x)
Therefore‖f − T3f‖Lp ≤ εDp‖f‖Lp
which shows that if we pick U small enough the operator T3 will be invertible.
This now means that any f ∈ Lp(G) ∗ Fn can be reconstructed in the following way (weonly write this up for the operator T2)
f =∑
i
cif(xi)T−12 (`xi
Fn)
or
f =∑
i
ci(T−12 f)(xi)`xi
Fn
The first representation in turn means that a v′ ∈ CouH∞n Lp(G) can be reconstructed from
the samples of its voice transform, i.e.
v′ =∑
i
ciWu(v′)(xi)W
−1u T−1`xi
Wu(u)
69
Remark 4.15. In [FG89a] it is concluded that the elements vi = W−1u T−1
2 `xiWu(u) are in
the space H1m. We cannot obtain such a result, since the space H1
m can be trivial in general(as for the case n = 2 for the Bergman spaces). We however claim, that the vi are in all thecoorbit spaces CouH∞n L
p(G) for np > 2 and so the same vectors can be used in all situations.
Remark 4.16. Note that we have avoided the use of integrability in the calculations above.This allows us to treat the case n = 2 which could not be treated in [FG88]. Further it mighthelp us when we try to generalize this construction of Bergman spaces to the whole scaleApα(D) for p ≥ 1 and α > 1 as coorbits of weighted Lp spaces on the group.
70
Chapter 5
Besov Spaces on Light Cones
The classical wavelet transform is related to the group R+ o R and the representationπ(a, b)f(x) = 1√
af(a−1(x−b)) which we met when discussing the Besov spaces in section 3.2.4
(in the case n = 1). We now replace R+ with another group, namely a group acting transi-tively on a symmetric cone. Square integrability and reproducing formulas have already beeninvestigated in this setting (see [BT96, FO03]). We construct a family of coorbit spaces for aspecial symmetric cone, and show that these are the Besov spaces introduced in [BBGR04].
5.1 Light Cones and Group Theory
5.1.1 Light Cones as Homogeneous Spaces
Let B(x, y) be the bilinear form on Rn given by
B(x, y) = xnyn − xn−1yn−1 − · · · − x1y1
and let SO0(n−1, 1) be the closed connected subgroup of GL(n,R) which leaves B invariant.The group SO0(n− 1, 1) has the Iwasawa decomposition ANK
A =
at =
cosh t 0 sinh t
0 In−2 0sinh t 0 cosh t
∣∣∣t ∈ R
N =
nv =
1 − |v|2/2 −vT |v|2/2v In−2 −v
−|v|2/2 −vT 1 + |v|2/2
∣∣∣v ∈ Rn−2
K =
kσ =
(σ 00 1
) ∣∣∣σ ∈ SO(n− 1)
where vT means the transpose of v.The forward light cone is the subset Λ of Rn satisfying
Λ = (x1, . . . , xn)|B(x, x) > 0, xn > 0
71
We can define the left action of an element λatnvkσ in the group R+SO0(n− 1, 1) by
(λatnvkσ).x = λ−1atnvkσx
and we wish to obtain a left-invariant measure on Λ in order to introduce the left-regularrepresentation of R+SO0(n− 1, 1) on L2(Λ)
`(λatnvkσ)f(x) = f((λatnvkσ)−1.w)
For this we need the determinant Det on the cone Λ given by
Det(x) =√B(x, x)
Then the measure Det(x)−n dx, where dx is the Lebesgue measure on Rn, is R+SO0(n−1, 1)-invariant as can be seen by these calculations
∫
Λ
f((λatnvkσ).x)dx
Det(x)n=
∫
Λ
f(λ−1atnvkσx)dx
Det(x)n
=
∫
Λ
f(x)λndx
Det(λ(atnvkσ)−1x)n
=
∫
Λ
f(x)dx
Det(x)n
The subgroup K leaves the base point e = (0, . . . , 0, 1) invariant and therefore the groupH = R+AN acts simply transitively on the forward light cone, i.e. every x ∈ Λ can bewritten x = (λatnv).e. To see this note that
atnv =
cosh t 0 sinh t0 In−2 0
sinh t 0 cosh t
1 − |v|2/2 −vT |v|2/2v In−2 −v
−|v|2/2 −vT 1 + |v|2/2
=
cosh t− et|v|2/2 −etvT sinh t+ et|v|2/2
v In−2 −vsinh t− et|v|2/2 −etvT cosh t+ et|v|2/2
so
(λatnv).e = λ−1
sinh t+ et|v|2/2
−vcosh t+ et|v|2/2
=
x1...xn
Noting that 0 < B(x, x) < x2n − x2
1 = (xn − x1)(xn + x1), so xn − x1 > 0 and xn + x1 > 0 wedetermine unique λ,t and v by
λ−1 = Det(x), v = −λ(x2, . . . , xn−1)T , t = − ln(λ(xn − x1)),
We have thus shown,that the forward light cone is a homogeneous space
Λ ' R+SO0(n− 1, 1)/K ' R+AN
72
The left invariant measure on H is given by
∫
H
f(h)dh =
∫
H
f(λatnv)dλ dv dt
λ
where dt,dv and dλ are the Lebesgue measures on R,Rn−2 and R+ respectively. Furthermorethe left invariant measure on the cone is given by
f 7→∫
Λ
f(x)dx
Det(x)n
so we can pass from an integral over the cone to an integral over the group by
∫
Λ
f(x)dx
Det(x)n=
∫
H
f((λatnv).e)dλ dv dt
λ
An integral over the light cone with respect to Lebesgue mesure can therefore be writen asan integral over the group in the following way
∫
Λ
f(x) dx =
∫
H
f((γatnv).e)dγ dv dt
γn+1
We now find the right Haar measure and the modular function on H . Notice that
atnva−1t =
cosh t 0 sinh t0 I 0
sinh t 0 cosh t
1 − |v|2/2 −vT |v|2/2v I −v
−|v|2/2 −vT 1 + |v|2/2
cosh t 0 − sinh t0 I 0
− sinh t 0 cosh t
=
1 − |etv|2/2 −etvT |etv|2/2
etv I −etv−|etv|2/2 −etvT 1 + |etv|2/2
= netv
The following calculations
∫f(γatnvλasnu)e
(n−2)tdγ dt dv
γ=
∫f(γatnvasnu)e
(n−2)tdγ dt dv
γ
=
∫f(γatnvnesuas)e
(n−2)tdγ dt dv
γ
=
∫f(γatnvas)e
(n−2)tdγ dt dv
γ
=
∫f(γatasne−sv)e
(n−2)tdγ dt dv
γ
=
∫f(γatasnv)e
(n−2)te(n−2)s dγ dt dv
γ
=
∫f(γatnv)e
(n−2)tdγ dt dv
γ
73
then establish that the right Haar measure on H is given by
f 7→∫
R+×R×Rn−2
f(γatnv)e(n−2)t dγ dt dv
γ
The modular function on H is then ∆(λasnu) = e(n−2)s satisfying∫
H
f(γatnvλasnu)dγ dt dv
γ= ∆(λasnu)
∫
H
f(γatnv)dγ dt dv
γ
and ∫
H
f((γatnv)−1)
dγ dt dv
γ=
∫
H
f(γatnv)∆(γatnv)−1dγ dt dv
γ
5.1.2 Fourier Transform on Light Cones
We now introduce the Fourier transform related to the bilinear form B:
f(w) = F(f)(w) =1√2π
n
∫
Rn
f(x)e−iB(x,w) dx
This Fourier transform is the usual Fourier transform in one dimension followed by an in-verse Fourier transform in n − 1 dimensions and therefore F inherits the properties of theusual Fourier transform. Thus we know that F is unitary on L2(Rn) and is a topologicalisomorphism from S(Rn) onto S(Rn). It acts on convolutions like the usual Fourier transform
f ∗ g(w) =√
2πnf(w)g(w)
as the following calculation shows (for f, g ∈ S(Rn))
f ∗ g(w) =1√2π
n
∫
Rn
f ∗ g(y)e−iB(y,w) dy
=1√2π
n
∫
Rn
(∫
Rn
f(x)g(y − x) dx)e−iB(y,w) dy
=1√2π
n
∫
Rn
f(x)(∫
Rn
g(y − x) e−iB(y,w) dy)dx
=1√2π
n
∫
Rn
f(x)(∫
Rn
g(y) dy e−iB(y+x,w) dy)dx
=√
2πn( 1√
2πn
∫
Rn
f(x)e−iB(x,w) dx)( 1√
2πn
∫
Rn
g(y) e−iB(y,w) dy)
=√
2πnf(w)g(w)
Since F : S(Rn) → S(Rn) is a topological isomorphism, we can extend the Fourier transformto tempered distributions in the usual way. In this text we work with the conjugate dual(S(Rn))∗ of S(Rn) (in order for it to resemble an inner product) and thus we define the
Fourier transform L for L ∈ (S(Rn))∗ by
〈L, φ〉 = 〈L, φ〉
One of the reasons for introducing this new Fourier transform is that F will be anintertwining operator between two natural representations of H o Rn.
74
5.2 Wavelets on Light Cones
Let the group G = H o Rn then this group has a natural representation on
L2Λ = f ∈ L2(Rn)|supp(f) ⊆ Λ
given by
π(λatnv, b)f(x) =1
λn/2f((λatnv)
−1(x− b))
This generalizes the quasi-regular representation of the group R+ o R from the classicalwavelet transform. In the Fourier domain this representation becomes
π(λatnv, b)f(w) = λn/2f(λ(atnv)−1w))e−iB(y,w)
and we recognize that it arises from the left action of H on the cone Λ, and that F is anintertwining operator. The group G has left invariant measure given by
∫f(g)dh =
∫f(λatnv, b)
dλ dv dt db
λn+1
The following result has a generalization to symmetric cones (see for example [FO03] and[BT96]) and ensures that wavelets for this representation exist.
Theorem 5.1. The representation (π, L2Λ) is square-integrable.
However we do not need this result and instead we will prove the properties neededfor construction of coorbit spaces. We first introduce the space SΛ of rapidly decreasingfunctions whose Fourier transform is supported on the cone, i.e.
SΛ = f ∈ S(Rn)|supp(f) ⊆ Λ
This space will be equipped with the subspace topology it inherits from S(Rn). The repre-sentation π can be restricted to SΛ and we denote the resulting representation by (π,SΛ) orsimply π.
Lemma 5.2. Let ψ ∈ SΛ be compactly supported such that 0 ≤ ψ ≤ 1 and 1/2 < ψ ≤ 1 ona neighborhood U of e, then ψ is cyclic in (π,SΛ).
Proof. The Fourier transform F has the same properties as the usual Fourier transform.The calculations below are immediate adaptations of results found in for example [Rud91,Chapter 6 and 7].
Let L be in the conjugate dual of SΛ and assume that 〈L, π(γatnv, b)ψ〉 = 0 for all(γatnv, b) ∈ G. Then the Fourier transform can be used to obtain
〈L, π(γatnv, b)ψ〉 = 0
Let eb(w) = e−iB(b,w) then the equation above can be rewritten to
0 = 〈L, ebπ(γatnv, 0)ψ〉 = 〈π(γatnv, 0)ψL, eb〉
75
which shows that the compactly supported functional π(γatnv, 0)ψL is equal to 0 (see [Rud91,
Theorem 7.23]). This means that for all φ ∈ SΛ for which φ has compact support C ⊆ Λ wehave the equalities
〈L, φπ(γatnv, 0)ψ〉 = 〈π(γatnv, 0)ψL, φ〉 = 0
We will now show that 〈ψ, L〉 is also 0. Since C is compact we can cover C by a finite numberof translates of U
C ⊆m⋃
i=1
(γatnv)i.U
Define the function
Ψ =m∑
i=1
π((γatnv)i, 0)ψ
which has support containing C (here we use that ψ is bounded away from 0 on U). Then
φ/Ψ is in C∞c and we see that
〈L, φ〉 = 〈ΨL, φ/Ψ〉 =n∑
i=1
〈π(γatnv, 0)ψL, φ/Ψ〉 = 0
Lastly any function in SΛ can be approximated by a function whose Fourier transform iscompact, and therefore L = 0 in the conjugate dual of SΛ.
Lemma 5.3. There is a non-zero constant Cψ such that the reproducing formula
Wψ(φ) ∗Wψ(ψ) = CψWψ(φ)
holds for φ ∈ SΛ. The constant Cψ is determined by
Cψ =
∫
Λ
|ψ(w)|2Det(w)2(1−n)(wn − w1)n−2 dw
Proof. Let eb(x) = e−iB(x,b) then
Wψ(φ) ∗Wψ(ψ)(λatnv, b)
=
∫
G
Wψ(φ)(γasnu, c)Wψ(ψ)((γasnu, c)−1(λatnv, b))
dγ ds du dc
γn+1
=
∫
G
(φ, π(γasnu, c)ψ)(π(γasnu, c)ψ, π(λatnv, b)ψ)dγ ds du dc
γn+1
=
∫
G
(φ, ecπ(γasnu, 0)ψ)(ecπ(γasnu, 0)ψ, ebπ(λatnv, 0)ψ)dγ ds du dc
γn+1
=
∫
G
(φ, ecπ(γasnu, 0)ψ)(π(γasnu, 0)ψ, eb−cπ(λatnv, 0)ψ)dγ ds du dc
γn+1
76
Let fγ,s,u = φπ(γasnu, 0)ψ and gγ,s,u = π(γasnu, 0)ψπ(λatnv, 0)ψ then the first inner productabove is fγ,s,u(b) and the second is gγ,s,u(c−b). Using the properties of the Fourier transformwe can continue our calculations
Wψ(φ) ∗Wψ(ψ)(λatnv, b) =
∫fγ,s,u ∗ gγ,s,u(b)
dγ ds du
γn+1
=
∫fγ,s,u(w)gγ,s,u(w)eiB(w,b) dw
dγ ds du
γn+1
=
∫φ(w)|π(γasnu, 0)ψ(w)|2π(λatnv, 0)ψ(w)eiB(w,b) dw
dγ ds du
γn+1
by use of Fubini’s theorem we can swap the order of integration
=
∫|π(γasnu, 0)ψ(w)|2dγ ds du
γn+1
∫φ(w)π(λatnv, b)ψ(w) dw
= Cψ(φ, π(λatnv, b)ψ)
= CψWψ(φ)(λatnv, b)
where
Cψ =
∫|π(γasnu, 0)ψ(w)|2dγ ds du
γn+1
We now show that Cψ is indeed a constant. Remember that any w ∈ Λ can be writtenuniquely as w = (σarnz).e, and thus we obtain
Cψ =
∫|π(γasnu, 0)ψ(w)|2dγ ds du
γn+1
=
∫|ψ((γasnu)
−1.(σarnz).e)|2dγ ds du
γ
=
∫|ψ(γσ−1n−ua−sarnze)|2
dγ ds du
γ
the invariance of dγγ
yields
=
∫|ψ(γn−uar−snze)|2
dγ ds du
γ
=
∫|ψ(γn−ua−snze)|2
dγ ds du
γ
and since a−snz = ne−sza−s the above expression is equal to
=
∫|ψ(γn−u+e−sza−se)|2
dγ ds du
γ
=
∫|ψ(γn−ua−se)|2
dγ ds du
γ
77
This shows that Cψ does not depend on w. Further we can rewrite
Cψ =
∫|ψ((γasnu)
−1.e)|2dγ ds duγ
=
∫|ψ((γasnu).e)|2∆(γasnu)
−1dγ ds du
γ
This can be rewritten to an Lebesgue integral over the cone, noting that ∆(γasnu)−1 =
Det(w)2−n(wn − w1)n−2. We get
Cψ =
∫
Λ
|ψ(w)|2Det(w)2(1−n)(wn − w1)n−2 dw
This finishes the proof.
Lemma 5.4. If f ∈ SΛ and k, l are non-negative integers then there is a constant Ck suchthat
|f(w)| ≤ Ck‖f‖k,lDet(w)k
(1 + |w|2)l
Proof. The function f vanishes at any point on the boundary of Λ. Therefore Taylor’stheorem tells us that f(w) around the point w0 ∈ ∂Λ closest to w vanishes faster than|w − w0|k for any k
f(w) =∑
|α|=kRα(w)(w − w0)
α
where the remainder Rα satisfies
|Rα(w)| ≤ 1
α!supw∈Λ
|∂αf(w)|
≤ 1
α!
supw∈Λ |(1 + |w|2)l∂αf(w)|(1 + |w|2)l
≤ 1
α!
‖f‖k,l(1 + |w|2)l
Since |(w − w0)α| ≤ |w − w0||α| it is apparent that
|f(w)| ≤ Ck‖f‖k,l|w − w0|k(1 + |w|2)l
We wish to get an estimate for the distance |w − w0|. Notice that any w ∈ Λ can bewritten w = γkate where k ∈ K. If w0 is the point closest to w then k−1w0 is closest toγate and therefore we reduce the problem to two dimensions (see figure). The point k−1w0
on the boundary has the form λ(1, 0, . . . , 0, 1) if t > 0 and thus the square of the distance
|w0 − w|2 = |λ(1, 0, . . . , 0, 1) − γate|2 = (λ− γ sinh t)2 + (λ− γ cosh t)2
78
x1x1x1x1x1x1
xnxnxnxnxnxn
eeeeee
w = γatnvew = γatnvew = γatnvew = γatnvew = γatnvew = γatnve
w0 = λ(1, 1)w0 = λ(1, 1)w0 = λ(1, 1)w0 = λ(1, 1)w0 = λ(1, 1)w0 = λ(1, 1)
Figure 5.1: Minimum distance from w ∈ Λ to w0 ∈ ∂Λ
is a function f(λ) which we minimize. We get f ′(λ) = 4λ− 2γet which is 0 if λ = γet/2. Inthis case
|w0 − w|2 = γ2e−2t/2 ≤ γ2/2 ≤ γ2 = Det(w)2
where it has been used that t > 0. We can carry out a similar analysis with t < 0 and obtainthe required estimate.
We will further need an estimate of the wavelet coefficients of Schwartz functions. Theestimate actually shows that the wavelet coefficients are integrable.
Lemma 5.5. The mapping
SΛ 3 φ 7→∫
G
|Wψ(φ)(γatnv, b)|γrdγ dt dv db
γn+1∈ C
is continuous for all r ∈ R.
Proof. First note that the wavelet coefficients can be rewritten as
Wψ(f)(γatnv, b) = (f, π(γatnv, b)ψ)
= γn/2∫
Λ
f(w)ψ(γn−va−tw)e−iB(w,b) dw
= −b−2i γn/2
∫
Λ
∂2
∂w2i
[f(w)ψ(γn−va−tw)]e−iB(w,b) dw
79
where we have used integration by parts twice. Therefore if we denote by L the LaplacianL = −
∑nk=1
∂2
∂w2k
we obtain
(1 + |b|2)Wψ(f)(γatnv, b) = γn/2∫
Λ
(1 + L)[f(w)ψ(γn−va−tw)]e−iB(w,b) dw
If we repeat the argument we are able to obtain
|Wψ(f)(γatnv, b)| = (1 + |b|2)−Nγn/2∫
Λ
|(1 + L)N [f(w)ψ(γn−va−tw)]| dw
for any N , thus proving that the wavelet coefficients are indeed integrable in b. We see that
(1 + L)N [f(w)ψ(γn−va−tw)] =∑
|α+β|≤2N
pβ(γn−va−t)∂αf(w)∂βψ(γn−va−tw)
where α, β are multi-indices and pβ(γn−va−t) are polynomials in the entries of γn−va−t. Wethus have to show the integrability in γ,t and v of expressions of the form
|pβ(γn−va−t)|γn/2∫
Λ
|∂αf(w)∂βψ(γn−va−tw)| dw
A change of variable and use of the fact that C = supp(ψ) is compact, reduces this to
γ−n/2|pβ(γn−va−t)|‖∂βψ‖∞∫
C
|∂αf(γ−1atnvw)| dw
By Lemma 5.4 we can estimate any such expression by
Ckγ−n/2|pβ(γn−va−t)|‖∂βψ‖∞‖∂αφ‖k,l
∫
C
Det(γ−1w)k
(1 + |γ−1atnvw|2)lγn/2 dw
for arbitrary k, l. Since the set C is compact, w is bounded away from 0 and we see that
C1(1 + |γ−1atnve|2) ≤ 1 + |γ−1atnvw|2 ≤ C2(1 + |γ−1atnve|2)
and we can estimate by
Ck‖∂βψ‖∞‖∂αφ‖k,l|pβ(γn−va−t)|γ−k−n/2
(1 + |γ−1atnve|2)l
All that is left now is to show
∫
H
|pβ(γn−va−t)|γ−k−n/2(1 + |γ−1atnve|2)l
γrdγ dt dv
γn+1<∞ (5.1)
We split this integral into two cases.
80
Case 1: 0 < γ ≤ 1The expression (5.1) can be estimated by
∫
Rn−2
∫
R
∫ 1
0
|pβ(n−va−t)|γr−k−3n/2−1
|γ−1atnve|2ldγ dt dv
=
∫
Rn−2
∫
R
∫ 1
0
|pβ(n−va−t)|γ2l+r−k−3n/2−1
|atnve|2ldγ dt dv
The integral over γ is finite for 0 < γ ≤ 1 if l is chosen large enough (and k = 0). Now
n−va−t =
cosh t− et|v|2 vT − sinh t+ et|v|2−etv I etv
− sinh t− et|v|2 vT cosh t+ et|v|2
and
atnve =
sinh t+ et|v|2−v
cosh t+ et|v|2
so we see that |pβ(n−va−t)| will be dominated by |atnve|2l ≥ 1 if l > deg(pβ). Thuschoosing l large enough the integral in (5.1) is finite.
Case 2: γ ≥ 1.In this case (5.1) can be estimated by
∫
Rn−2
∫
R
|pβ(n−va−t)||atnve|2l
dt dv
∫ ∞
1
γr+2l+deg(pβ)−k−3n/2−1 dγ
The first integral is finite if l is large enough, and the second integral is finite when kis chosen large enough (depending on l).
To sum up we have obtained the following estimate∫
G
|Wψ(φ)(γatnv, b)|γrdγ dt dv db
γn+1≤ C
∑
|α+β|≤2N
‖∂βψ‖∞‖∂αφ‖k,l
which shows the continuous dependence on φ.
Denote by Lp,qs (G) the space of measurable functions f on the group for which
‖f‖Lp,qs
=(∫
H
( ∫
Rn
|f(γatnv, b)|p db)q/p dγ dt dv
γn+1
)1/q
<∞
then the integrability of Wψ(ψ) shows that
Lemma 5.6. Lp,qs ∗Wψ(ψ) ⊆ Lp,qs and
Lp,qs 3 F 7→ F ∗Wψ(ψ) ∈ Lp,qs
is continuous.
81
Further the integrability also shows that for 1/p+1/p′ = 1 and 1/q+1/q′ = 1 the wavelet
coefficient is in Lp′,q′
1/s and therefore
Lemma 5.7. The mapping
SΛ 3 φ 7→∫
G
F (x)Wψ(φ)(x) dx ∈ C
is continuous for all F ∈ Lp,qs .
This verifies the assumptions for construction of coorbit spaces for the spaces Lp,qs andtherefore we can define
CoψSΛLp,qs = Φ ∈ S ′
Λ|Wψ(Φ) ∈ Lp,qs In the next section we will show that these coorbit spaces are the Besov spaces inttroducedin [BBGR04].
Remark 5.8 (Discretization). The representation used for this construction is integrable(as we have shown) and therefore the discretization procedure by Feichtinger and Grochenigcan be used directly.
5.3 Littlewood-Paley Decomposition and Besov Spaces
on the Light Cone
In this section we introduce a Littlewood-Paley decomposition of the forward light cone.This decomposition can be used to analyse pseudo-differential operators and to define Besovspaces on the cone. The decomposition has been carried out for all symmetric cones in[BBGR04] and we refer to this article for proofs. We then present the last result of thisthesis, namely a wavelet description of the Besov spaces, as we show they correspond to thecoorbit spaces defined in the previous section.
The group R+A is an abelian group with exponential function exp : R×R → R+A givenby
exp(t, s) = et
sinh s0
cosh s
Let Vr = (s, t) ∈ R × R|s2 + t2 < r define the K-invariant ball Br(e) = K exp(Vr). Thenfor w = γatnve ∈ Λ we define the ball of radius r centered at w to be
Br(w) = γatnvBr(e)
The following covering lemma for the cone can be extracted from Lemma 2.6 in [BBGR04]and is illustrated in Figure 5.2.
Lemma 5.9 (Whitney cover with lattice points wj). Given δ > 0 there exists a sequencewj ⊆ Λ such that Bδ/2(wj) are disjoint and Bδ(wj) cover Λ with the property that there isan N such that w ∈ Λ belongs to at most N of the balls Bδ(wj) (finite intersection property).
82
eeeee
Br(e)Br(e)Br(e)Br(e)Br(e)
(a) The ball Br(e)
Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)Br(wj)
Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)Br(wj−1)
Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)Br(wj−2)
Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)Br(wj+1)
Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)Br(wj+2)
(b) Translates of Br(e)
(c) Cover with translates of Br(e) (d) Translates of Br/2(e) have no overlap
Figure 5.2: Covering of the cone
We now see how to construct a smooth partition of unity subordinate to a cover fromLemma 5.9. Let 0 ≤ ϕ ≤ 1 be a smooth function with support in B2δ(e) such that ϕ = 1on Bδ(e). Each of the points wj ∈ Λ can be written wj = γ−1
j atjnvje for gj = γ−1
j atjnvj∈
R+AN and now we define ϕj(w) = ϕ(g−1j w). Then the function Φ =
∑j ϕj is smooth
and bounded from above and below (by the finite intersection property), and we can finally
define ψj = ϕj/Φ. We then see that ψj is smooth and with compact support in B2δ(wj),
ψj = 1 on Bδ/2(wj) and∑
j ψj(w) = 1 for all w ∈ Λ. Such a partition of unity is called aLittlewood-Paley decomposition of the cone subordinate to a Whitney cover. We are nowready to define the Besov spaces on the light cone as in [BBGR04]
Definition 5.10. Let ψj be a Littlewood-Paley decomposition of the cone subordinate to aWhitney cover with lattice points wj . For 1 ≤ p, q <∞ define the norm
‖f‖Bp,qs
=(∑
j
Det−s(wj)‖f ∗ ψj‖qp)1/q
then the space Bp,qs consist of the f ∈ S ′
Λ for which ‖f‖Bp,qs<∞.
In [BBGR04, Lemma 3.8] it is further proven, that Bp,qs does not depend (up to norm
83
equivalence) on the functions ψj nor on the Whitney decomposition. We will use this in thesequel.
The main result of this chapter is that the coorbits defined in the end of section 5.2 arein fact Besov spaces.
Theorem 5.11. The Besov spaces can be described as coorbits, Bp,qn−s−nq/2 = CoψSΛ
Lp,qs (G),with equivalent norms.
Proof. We first show that Bp,qs+nq/2−n ⊆ CoψSΛ
Lp,qs (G): Assume that f ∈ Bp,qs+nq/2−n and that
φi is a Littlewood-Paley decomposition of the cone with lattice points wi = gie = γ−1i atinvi
e.
Let ψ be a wavelet for which ψ has compact support in a neighbourhood of U of e. By ψγatnv
denote the functionψγatnv(x) = γ−nψ(−(γatnv)−1x)
then
Wψ(f)(γatnv, b) = γ−n/2∫f(x)ψ((γatnv)−1(x− b)) dx = γn/2f ∗ ψγatnv(b)
Let the disjoint sets Vi ⊆ Λ cover Λ and satisfy Vi ⊆ giU . Now choose the subsets Ui ofthe group H such that γatnv ∈ Ui if γ−1atnve ∈ Vi We can then write the Lp,qs norm of thewavelet coefficient as
‖Wψ(f)‖Lp,qs
=(∫
Λ
γs+nq/2‖f ∗ ψγatnv‖qpdγ dt dv
γn+1
)1/q
≤(∑
i
∫
Ui
γs+nq/2−n‖f ∗ ψγatnv‖qpdγ dt dv
γ
)1/q
≤ C(∑
i
∫
Ui
γs+nq/2−ni ‖f ∗ ψγatnv‖qp
dγ dt dv
γ
)1/q
where we have used that γ is comparable to γi = Det(wi)−1 inside the set Ui. For any j
define φi,j = `gjφi. Since φii is a Littlewood-Paley decomposition of the cone the systems
φi,jj (with index j) and φi,ji (with index i), also for Littlewood-Paley decompositionsof the cone. For fixed i we thus can write ‖f ∗ ψγatnv‖p as
‖f ∗ ψγatnv‖p =∥∥∥
∑
j∈Jf ∗ ψγatnv ∗ φi,j
∥∥∥p≤
∑
j∈J‖f ∗ ψγatnv ∗ φi,j‖p
The index set J in this sum is finite, since both ψ and φ are compactly supported and wiare well-spread. Furhter the index set J can be chosen large enough that it neither dependson i nor on γ−1atnv ∈ Ui. The L1(Rn) norm of ψγatnv is uniformly bounded from above, infact ‖ψγatnv‖L1(Rn) = ‖ψ‖L1(Rn), so we obtain that
‖f ∗ ψγatnv‖p ≤∑
j∈J‖f ∗ φi,j‖p
84
Inserting this in the inequalities above and using that Ui ⊆ giU has uniformly boundedmeasure (since dγ dt dv
γis the invariant measure on H) we get
‖f‖Lp,qs
≤(∑
i
∫
Ui
γs+nq/2−ni
(∑
j∈J‖f ∗ φi,j‖p
)q dγ dt dvγ
)1/q
≤ C∑
j∈J
(∑
i
γs+nq/2−ni ‖f ∗ φi,j‖qp
)1/q
where we also applied the triangle inequality for the `q-norm. When we translate φi by gj to
get a new Littlewood-Paley decomposition φi,ji the associated Besov space norm is givenby
‖f‖Bp,qs
=(∑
i
Det−s(gjwi)‖f ∗ ψi,j‖qp)1/q
Now set γi,j = Det(gjwi)−1 = γiγj, then, since the sum over J is finite, any γi is comparable
to γi,j for each j. Finally we obtain
‖f‖Lp,qs
≤ C∑
j∈J
(∑
i
γs+nq/2−ni ‖f ∗ φi,j‖qp
)1/q
≤ C∑
j∈J
(∑
i
γs+nq/2−ni,j ‖f ∗ φi,j‖qp
)1/q
Since each of the φi,ji form a decomposition of the cone, each of the terms
(∑
i
γs+nq/2−ni,j ‖f ∗ φi,j‖qp
)1/q
=(∑
i
Det(gjwi)n−s−nq/2‖f ∗ φi,j‖qp
)1/q
is a Besov space norm and therefore comparable to ‖f‖Bp,qs+nq/2−n
(see [BBGR04, Lemma 3.8
and expression (3.20)]). This shows that there is a C > 0 such that
‖Wψ(f)‖Lp,qs
≤ C‖f‖Bp,qn−s−nq/2
Now let us show that CoψSΛLp,qs (G) ⊆ Bp,q
s+nq/2−n: Let φ be the smooth function with support
in B2δ(e) used to generate a Littlewood-Paley decomposition. The coorbit spaces are inde-
pendent of the wavelet ψ, so we ψ and a compact neighbourhood U ⊆ H such that Usupp(φ)
is contained in ψ−1(1) and the giU ’s have finite overlap (the gi’s come from the lattice
points wi = gie). This means that supp(φi) is contained in (ψγatnv)−1(1) for γ−1atnv ∈ giU .
Therefore φiψγatnv = φi for all γ−1atnv ∈ giU . We exploit this to see that
‖f ∗ φi‖qp =1
|U |
∫
giU
‖f ∗ φi‖qpdγ dt dv
γ
=1
|U |
∫
giU
‖f ∗ φi ∗ ψγatnv‖qpdγ dt dv
γ
≤ C
∫
giU
‖f ∗ ψγatnv‖qpdγ dt dv
γ
85
where dγ dt dvγ
is the invariant measure on the groupH . In the last step we used that ‖φi‖L1(Rn)
is uniformly bounded (see [BBGR04, Proposition 3.2(3)]). Inside each of the sets giU theγi are comparable to γ, and the sets giU overlap a finite amount of times, so we obtain theestimate ∑
i
γs+nq/2−ni ‖f ∗ φi‖qp ≤ C
∫
H
γs+nq/2−n‖f ∗ ψγatnv‖qpdγ dt dv
γ
We use this to find the estimate of the Besov space norm
‖f‖Bp,qs+nq/2−n
=(∑
i
Det(wi)n−s−nq/2‖f ∗ φi‖qp
)1/q
=(∑
i
γs+nq/2−ni ‖f ∗ φi‖qp
)1/q
≤ C(∫
H
γs+nq/2‖f ∗ ψγatnv‖qpdγ dt dv
γn+1
)
= C‖Wψ(f)‖Lp,qs
This proves the equivalence of the norms of the two spaces.
Remark 5.12. The wavelet characterization of Besov spaces on forward light cones seemsto generalize to all symmetric cones. We will deal with this in future work.
86
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Vita
Jens Gerlach Christensen was born in November 1975, in Hillerød, Denmark. In May 2003he finished his bachelor of science degree and master of science degree at University ofCopenhagen, Denmark. In August 2004 he came to Louisiana State University to pursuegraduate studies in mathematics. He is currently a candidate of Doctor of Philosophy inmathematics, which will be awarded in August 2009.
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