Lecture 8 – Wavelets in Functional Analysis

David Walnut Department of Mathematical Sciences George Mason University Fairfax, VA USA Chapman Lectures, Chapman University, Orange, CA

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Outline

Modulation spaces

The Feichtinger Algebra S0(R) Pseudodifferential operators and Gabor frames Wavelets as unconditional bases for Banach spaces Wavelets and operators

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Motivation

Suppose we are given a function f (x). How can we measure the time-frequency concentration of f ? Given g, ↵,> 0, what do the Gabor coefficients

c = f , T M g k,n h ↵k n i tell us about the smoothness and decay properties of f ? If we can write

f = f , T M g T M h ↵k n i ↵k n Xk,n for some analysis window g and synthesis window , what sort of smoothness and decay properties can g and share?

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Modulation Spaces

Definition (Short Time Fourier Transform) Given g L2(Rd ), we define the short-time Fourier transform 2 (STFT) on L2(Rd ) by

2⇡i(t ) Vgf (x,)= f (t) g(t x) e · dt = f , MTx g . d h i ZR

Definition (Modulation Space) Let g (R) 0 , and 1 p, q . The modulation spaace p,q 2S \{ }  1 M (R) consists of all f 0(R) such that 2S q/p 1/q p f p,q = V (x,!) dx d! k kM | g | ✓ ZR ✓ ZR ◆ ◆ and the obvious changes being made when p = or q = . 1 1

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis f Mp,q if and only if V f is in a so-called mixed-norm 2 g space, where for all !,

p Vg( ,!) L (R) · 2 and q Vg( ,!) p L (R). k · k 2 Intuitively, p measures the decay of f at infinity since

V f (x,!) ( f g )(x) | g | | |⇤| | so that if f Lp, V f ( ,!) Lp for all !. 2 | g · |2 q measures the smoothness of f in the sense that

V f (x,!)=(f\T g)(!) g · x will on average be in Lq for each x.

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis For 1 p , Mp,q(R) is a Banach space.  1 For 1 p < , the dual space of Mp,q(R) is identified with  1 the modulation space Mp0,q0 (R) where

1/p + 1/p0 = 1/q + 1/q0 = 1.

Mp,q(R) is independent of the window g (R) in the 2S sense that if another such window is used, the norms generated are equivalent. Mp,q(R) is invariant under time and frequency shifts, and f Mp,q(R) if and only if f Mq,p(R). 2 2 b

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Feichtinger Algebra

1,1 The modulation space M (R)=S0(R) is called the Feichtinger Algebra (Feichtinger, 1981).

S0(R) is the smallest Banach space invariant under time frequency shifts and under the Fourier transform. 1,1 , This makes S0 = M and its dual (S0)⇤ = M1 1 ideal substitutes for the Schwartz functions (R) and the S tempered distributions 0(R) in many instances. S S0(R) is a Banach algebra under pointwise multiplication and convolution.

S0(R) is the largest Banach space on which the Poisson Summation Formula holds pointwise.

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis S0 and Banach frames

Definition 2 Given g L (R) and ↵,> 0, the Gabor frame operator Sg,g is 2 defined by

S f = f , T M g T M g. g,g h ↵k n i ↵k n k Z n Z X2 X2

We denote the collection T↵k Mng : k, n Z by (g,↵,). { 2 2 } G Recall that Sg,g is an isomorphism of L (R) if and only if the collection (g,↵,) is a frame for L2(R). G Choosing g from M1,1(R) turns out to be the right choice of window class for Gabor frames.

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Sg,g on S0

Theorem

If g S0(R), then the following are equivalent. 2 1,1 (1) Sg,g is invertible on M (R). p,q (2) Sg,g is invertible on all of the modulation spaces M (R), 1 p .  1 In this case, (g,↵,) is a frame for L2(R) and the dual window 1,1 G M (R) as well. 2 This theorem allows us to move toward the notion of a Banach frame for Mp,q The idea is to characterize membership of f Mp,q by 2 some condition on the Gabor coefficients

f , T M g {h ↵k n i} where the window g M1,1. 2 Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Banach frames

Definition (Gröchenig, 1991)

A sequence en : n N in a Banach space B is called a { 2 } Banach frame if there exists an associated sequence space Bd (N), a constant C > 0, and a continuous operator R : B B such that for all f B, d ! 2 1 (1) f f , e C f , and C k kB  kh nikBd  k kB (2) R( f , e )=f . h ni Definition (Discrete mixed-norm spaces) For 1 p , define `p,q to be the space of sequences  1 a =(ak,n)k,n Z, for which 2 q/p 1/q p a p,q = a k k` | k,n| ✓ n Z ✓ k Z ◆ ◆ X2 X2

is finite. Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Theorem Assume that (g,↵,) is a frame for L2(R) with ↵ Q and G 2 g M1,1 Then there exists C > 0 such that for all 1 p , 2  1 and f Mp,q, 2 q/p 1/q 1 p f p,q f , T M g f p,q . C k kM  |h ↵k n i| k kM ✓ n Z ✓ k Z ◆ ◆ X2 X2 Moreover, there exists M1,1 such that f Mp,q can be 2 2 recovered by

f = f , T M g T M h ↵k n i ↵k n n Z k Z X2 X2 where the series converges unconditionally in Mp,q if 1 p, q < and weakly if p or q = .  1 1

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Pseudodifferential operators

Definition Let be a function or distribution on R2d . Then the operator

2⇡i(x !) Kf (x)= (x,!)f (!) e · d! d ZR is the pseudodifferential operator withb symbol .

Pseudodifferential operators arose in the mid-1960s and were formally described by Kohn and Nirenberg, 1965. The notion arose earlier in a different context in the study of time-varying communication channels by Zadeh, 1950.

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Motivation from PDE

Consider the N-th order differential operator with nonconstant coefficients given by

↵ Af (x)= a↵(x) D f (x). ↵ N | X| By Fourier inversion,

↵ ↵ 2⇡i(x !) D f (x)= (2⇡i!) f (!) e · d!. d ZR b ↵ 2⇡i(x !) Af (x)= a↵(x)(2⇡i!) f (!) e · d! d ZR ✓ ↵ N ◆ | X| which is the pseudodifferential operatorb with symbol

↵ (x,!)= a↵(x)(2⇡i!) ↵ N | X|

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Motivation from Communications

The standard model for a time-invariant communication channel is convolution.

Hf (x)= h(t) f (x t) dt. ZR The impulse response h completely characterizes the channel and does not change with time. In mobile communications, the impulse response can change with time, so the general model is

Hf (x)= h(x, t) f (x t) dt. ZR

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis 2⇡i!t Letting (x,!)= h(x, t) e dt, then by Fourier R inversion Z h(x, t)= (x,!) e2⇡i!t d!. ZR Substituting gives

Hf (x)= (x,!) e2⇡i!t f (x t) d! dt ZR ZR 2⇡i!x 2⇡i!t = (x,!) e f (t) e dt d! ZR ZR = (x,!) e2⇡i!x f (!) d!. ZR b

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis 2⇡i⌫x Letting ⌘(t,⌫)= h(x, t) e dx, then by Fourier R inversion Z h(x, t)= ⌘(t,⌫) e2⇡i⌫x d⌫. ZR Substituting gives

Hf (x)= ⌘(t,⌫) e2⇡i⌫x f (x t) d⌫ dt ZR ZR = ⌘(t,⌫) M⌫ Tt f (x) dt d⌫ ZR ZR so that H is realized as a superposition of time delays and Doppler shifts.

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis ⌘(t,⌫) is called the spreading function of the operator H and measures how much a delta impulse is “spread” in time and how a pure tone is “spread” in frequency. Note that

2⇡i⌫x ⌘(t,⌫)= h(x, t) e dx ZR 2⇡i!t 2⇡i⌫x = (x,!) e e d! dx ZR ZR 2⇡i(⌫x !t) = (x,!) e d! dx ZR ZR so that the spreading function is the symplectic Fourier transform of the symbol.

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis All of this suggests that Gabor analysis is a natural setting for studying the properties of pseudodifferential operators. An illustration of this is the following generalization of the Calderón-Vaillancourt theorem on the L2-boundedness of pseudodifferential operators.

Theorem (Calderón-Vaillancourt, 1971) Given a smooth symbol with bounded derivatives up to order 2d + 1, the pseudodifferential operator with symbol is bounded on L2(Rd ).

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis Theorem (Gröchenig and Heil, 1999) ,1 d If M1 (R ), then the pseudodifferential operator with 2 symbol is bounded on Mp,q(Rd ) for all 1 p, q , with  1 uniform bound K op C ,1 . k k  k kM1 2 d 2,2 d In particular, K is bounded on L (R )=M (R ).

Since the space of smooth symbols with bounded ,1 derivatives up to order 2d + 1 is embedded in M1 , this result is a generalization of the C-V Theorem. Note that the result assumes no smoothness on .

Walnut (GMU) Lecture 8 – Wavelets in Functional Analysis