Numerical Harmonic Analysis Group

FOURIER STANDARD SPACES and the Kernel Theorem

Hans G. Feichtinger [email protected] www.nuhag.eu . Currently Guest Prof. at TUM (with H. Boche)

Garching, TUM, July 13th, 2017

Hans G. Feichtinger [email protected] www.nuhag.euFOURIER. Currently STANDARD GuestSPACES Prof. at TUM and the (with Kernel H. Boche) Theorem OVERVIEW

d We will concentrate on the setting of the LCA group G = R , although all the results are valid in the setting of general locally compact Abelian groups as promoted by A. Weil. —————————————————————- Classical Fourier Analysis pays a lot of attention to p d
L (R ), k · kp because these spaces (specifically for p ∈ {1, 2, ∞}) are important to set up the Fourier transform as an integral transform which also respects convolution (we have the convolution theorem) and preserving the energy (meaning that it is 2 d
a unitary transform of the Hilbert space L (R ), k · k2 ). —————————————————————- d Occasionally the Schwartz space S(R ) is used and its dual 0 d S (R ), the space of tempered distributions (e.g. for PDE and d the kernel theorem, identifying operators from S(R ) to 0 d 0 2d S (R ) with their distributional kernels in S (R )).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem OVERVIEW II

S d
In the last 2-3 decades the Segal algebra 0(R ), k · kS0 1 d (equal to the modulation space (M (R ), k · kM1 )) and its dual, 0 d ∞ d S 0 M ( 0 (R ), k · kS0 ) or (R ) have gained importance for many questions of Gabor analysis or time-frequency analysis. Fourier standard spaces is a new name for a class of Banach d 0 d spaces sandwiched in between S0(R ) and S0 (R ), with two module structures, one with respect to the Banach convolution 1 d
algebra L (R ), k · k1 , and the other by pointwise multiplication L1 d
with elements of the Fourier algebra F (R ), k · kFL1 . As we shall point out there is a huge variety of such spaces, and many questions of Fourier analysis find an appropriate description in this context.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem OVERVIEW III

The spaces in this family are useful for a discussion of questions in Gabor Analysis, which is an important branch of time-frequency analysis, but also for problems of classical Fourier Analysis, such as the discussion of Fourier multipliers, Fourier inversion questions (requiring to work 1 d 1 d with the space L (R ) ∩ FL (R )), and many other spaces. Within the family there are two subfamilies, namely the Wiener amalgam spaces and the so-called modulation spaces, among them S d
the Segal algebra 0(R ), k · kS0 or Wiener’s algebra 1 d
W (C0, ` )(R ), k · kW .

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The key-players for time-frequency analysis

Time-shifts and Frequency shifts

Tx f (t) = f (t − x) d and x, ω, t ∈ R 2πiω·t Mωf (t) = e f (t) . Behavior under Fourier transform ˆ ˆ (Tx f )b= M−x f (Mωf )b= Tωf

The Short-Time Fourier Transform

Vg f (λ) = hf , MωTt gi = hf ,π (λ)gi = hf , gλi, λ = (t, ω);

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem A Typical Musical STFT

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Demonstration using GEOGEBRA (very easy to use!!)

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Spectrogramm versus Gabor Analysis

Assuming that we use as a “window” a Schwartz function d 2 g ∈ S(R ), or even the Gauss function g0(t) = exp(−π|t| ), we can define the spectrogram for general tempered distributions 0 d f ∈ S (R )! It is a continuous function over phase space. In fact, for the case of the Gauss function it is analytic and in fact a member of the Fock space, of interest within complex analysis.

Both from a pratical point of view and in view of this good smoothness one may expect that it is enough to sample this spectrogram, denoted by Vg (f ) and still be able to reconstruct f (in analogy to the reconstruction of a band-limited signal from regular samples, according to Shannon’s theorem).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem So let us start from the continuous spectrogram

2 d The spectrogram Vg (f ), with g, f ∈ L (R ) is well defined and has a number of good properties. Cauchy-Schwarz implies:

2 d kVg (f )k∞ ≤ kf k2kgk2, f , g ∈ L (R ),

d d in fact Vg (f ) ∈ C0(R × Rb ). Plancherel’s Theorem gives

2 d kVg (f )k2 = kgk2kf k2, g, f ∈ L (R ).

2 d Assuming that g is normalized in L (R ), or kgk2 = 1 makes f 7→ Vg (f ) isometric, hence we request this from now on. Note: Vg (f ) is a complex-valued function, so we usually look 2 at |Vg (f )|, or perhaps better |Vg (f )| , which can be viewed as d d a probability distribution over R × Rb if kf k2 = 1 = kgk2.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The continuous reconstruction formula

Now we can apply a simple abstract principle: Given an isometric embedding T of H1 into H2 the inverse (in the range) is given by ∗ the adjoint operator T : H2 → H1, simply because ∀h ∈ H1 2 2 ∗ hh, hiH1 = khkH1 = (!) kThkH2 = hTh, ThiH2 = hh, T ThiH1 , (1) and thus by the polarization principle T ∗T = Id. 2 d In our setting we have (assuming kgk2 = 1) H1 = L (R ) and 2 d d H2 = L (R × Rb ), and T = Vg . It is easy to check that Z ∗ L2 d d Vg (F ) = F (λ)π(λ)g dλ, F ∈ (R × Rb ), (2) Rd ×Rbd 2 d understood in the weak sense, i.e. for h ∈ L (R ) we expect: Z ∗ hV (F ), hi 2 = F (x) · hπ(λ)g, hi 2 dλ. (3) g L (Rd ) L (Rd ) Rd ×Rbd

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Continuous reconstruction formula II

Putting things together we have Z ∗ hf , hi = hVg (Vg (f )), hi = Vg (f )(λ) · Vg (h)(λ) dλ. (4) Rd ×Rbd

A more suggestive presentation uses the symbol gλ := π(λ)g and describes the inversion formula for kgk2 = 1 as: Z 2 d f = hf , gλi gλ dλ, f ∈ L (R ). (5) Rd ×Rbd This is quite analogous to the situation of the Fourier transform Z 2 d f = hf , χs i χs ds, f ∈ L (R ), (6) Rd d with χs (t) = exp(2πihs, ti), t, s ∈ R , describing the “pure d frequencies” (plane waves, resp. characters of R ). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Discretizing the continuous reconstruction formula

Note the crucial difference between the classical formula (6) (Fourier inversion) and the new formula formula (5). The building 2 d blocks gλ belong to the Hilbert space L (R ), in contrast to the 2 d characters χs ∈/ L (R ). Hence finite partial sums cannot 2 d approximate the functions f ∈ L (R ) in the Fourier case, but they 2 d can (and in fact do) approximate f in the L (R )-sense. The continuous reconstruction formula suggests that sufficiently fine (and extended) Riemannian-sum-type expressions approximate f . This is a valid view-point, at least for nice windows g (any Schwartz function, or any classical summability kernel is OK: see [F. Weisz] Inversion of the short-time Fourier transform using Riemannian sums for example [7]).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem S d S 0 d Modulation spaces, in particular 0(R ) and 0 (R )

The reconstruction of f from its STFT (Short-time Fourier Transform) suggests that at least for “good windows” g one can control the smoothness (and/or decay) of a function or distribution by controlling the decay of Vg (f ) in the frequency resp. the time direction. A polynomial weight depending on the frequency variable only can be used to describe Sobolev spaces, and (weighted) mixed-norm conditions can be used to define the (now classical) modulation s d
spaces M ( ), k · kMs . p,q R p,q We will put particular emphasis on the modulation spaces d 1,1 1 S0(R ) = M = M , characterized by the membership of 1 2d 0 d ∞,∞ ∞ Vg (f ) ∈ L (R ) and S0 (R ) = M = M , with uniform 0 d convergence describing norm convergence in S0 (R ), while pointwise convergence corresponds to the w ∗-convergence in 0 d S0 (R ).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Banach Module Terminology

Definition

A Banach space (B, k · kB ) is a Banach module over a Banach algebra (A, ·, k · kA) if one has a bilinear mapping (a, b) 7→ a • b, from A × B into B bilinear and associative, such that

ka • bkB ≤ kakAkbkB ∀ a ∈ A, b ∈ B, (7)

a1 • (a2 • b) = (a1 · a2) • b ∀a1, a2 ∈ A, b ∈ B. (8)

Definition

A Banach space (B, k · kB ) is a Banach ideal in (or within, or of) a Banach algebra (A, ·, k · kA) if (B, k · kB ) is continuously embedded into (A, ·, k · kA), and if in addition (7) is valid with respect to the internal multiplication inherited from A.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Wendel’s Theorem

Theorem

1 1 1 The space of HL1 (L , L ) all bounded linear operators on L (G) which commute with translations (or equivalently: with convolutions) is naturally and isometrically identified with M ( b(G), k · kMb ). In terms of our formulas this means

1 1 d d 1 L L M HL ( , )(R ) ' ( b(R ), k · kMb ),

1 d via T ' Cµ : f 7→ µ ∗ f , f ∈ L , µ ∈ Mb(R ).

Lemma

BL1 = {f ∈ B | kTx f − f kB → 0, forx → 0}. d 1 d Consequently we have (Mb(R ))L1 = L (R ), the closed ideal of d absolutely continuous bounded measures on R .

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Pointwise Multipliers

Via the Fourier transform we have similar statements for the Fourier algebra, involving the Fourier Stieltjes algebra.

L1 L1 M d M d L1 HFL1 (F , F ) = F( b(R )), F( b(R ))FL1 = F . (9)

Theorem

d
The completion of C0(R ), k · k∞ (viewed as a Banach algebra and module over itself) is given by

C C C d
HC0 ( 0, 0) = b(R ), k · k∞ . C d C d On the other hand we have ( b(R ))C0 = 0(R ).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Essential part and closure

In the sequel we assume that (A, k · kA) is a Banach algebra with 1 d
bounded approximate units, such as L (R ), k · k1 (with C d
L1 d
convolution), or 0(R ), k · k∞ or F (R ), k · kFL1 with pointwise multiplication. Theorem Let A be a Banach algebra with bounded approximate units, and B a Banach module over A. Then we have the following general identifications:

A A A A A A (BA)A = BA, (B )A = BA, (BA) = B , (B ) = B . (10) or in a slightly more compact form:

A A A AA A BAA = BA, B A = BA, BA = B , B = B . (11)

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Essential Banach modules and BAIs

The usual way to define the essential part BA resp. Be of a Banach module (B, k · kB ) with respect to some Banach algebra action (a, b) 7→ a • b is defined as the closed linear span of A • B
within B, k · kB . This subspace has other nice characterizations using BAIs (bounded approximate units (BAI) in (A, k · kA)): Lemma

For any BAI (eα)α∈I in (A, k · kA) one has:

BA = {b ∈ B | lim eα • b = b} (12) α

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The Cohen-Hewitt Factorization Theorem

In particular one has: Let (eα)α∈I and (ub)β∈J be two bounded approximate units (i.e. bounded nets within (A, k · kA) acting in the limit like an identity in the Banach algebra (A, k · kA). Then

lim eα • b = b ⇔ lim uβ • b = b. (13) α β

Theorem (The Cohen-Hewitt factorization theorem, without proof, see [5]) Let (A, k · kA) be a Banach algebra with some BAI of size C > 0, then the algebra factorizes, which means that for every a ∈ A there exists a pair a0, h0 ∈ A such that a = h0 · a0, in short: A = A · A. In fact, one can even choose ka − a0k ≤ ε and kh0k ≤ C.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Essential part and closure II

Having now Banach spaces of distributions which have two module structures, we have to use corresponding symbols. FROM NOW ON we will use the letter A mostly for pointwise Banach algebras 1 and thus for the FL -action on (B, k · kB ), and we will use the symbol G (because convolution is coming from the integrated group action!) for the L1 convolution structure. We thus have G G G GG G BGG = BG , B G = BG , BG = B , B = B . (14) In this way we can combine the two operators (in view of the above formulas we can call them interior and closure operation) with respect to the two module actions and form spaces such as

G G G G B A, BA A, B A A ... or changes of arbitrary length, as long as the symbols A and G appear in alternating form (at any position, upper or lower).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Combining the two module structures

Fortunately one can verify (paper with W.Braun from 1983, J.Funct.Anal.) that any “long” chain can be reduced to a chain of at most two symbols, the last occurence of each of the two symbols being the relevant one! So in fact all the three symbols in the above chain describe the same space of distributions. But still we are left with the follwoing collection of altogether eight two-letter symbols:

G G A A AG GA BGA, BAG , BA , B A, BG , B G , B , B (15)

and of course the four one-symbol objects

A G BA, BG , B , B (16)

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Some structures, simple facts

There are other, quite simple and useful facts, such as

1 2 1 2 HA(B A, B ) = HA(B A, B A) (17)

1 1 which can easily be verified if B A = A • B , since then 1 2 1 10 T ∈ HA(B A, B ) applied to b = a • b gives

1 10 10 2 T (b ) = T (a • b ) = a • T (b ) ∈ B A.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The Main Diagram

This diagram is taken from [1].

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Fourier Standard Spaces, FouSS

Definition

A Banach space (B, k · kB ), continuously embedded between 0
S S 0 0(G) and 0 (G), k · kS0 , i.e. with

0
S B S 0 0(G), k · kS0 ,→ ( , k · kB ) ,→ 0 (G), k · kS0

is called a Fourier Standard Space on G (FSS of FoSS) if it has M a double module structure: over ( b(G), k · kMb ) with respect to convolution and over (the Fourier-Stieltjes algebra) F(Mb(Gb)) with respect to pointwise multiplication.

REMARK: One could unify this assumption by combining the two separate (commutative) group actions by the integrated d d group action of the reduced Heisenberg group R × R × T under the Schr¨odingerrepresentation: π(t, s, τ) = τMs Tt .

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem TF-homogeneous Banach Spaces

A sufficient setting is the following one: Definition

A Banach space (B, k · kB ) with

d 0 d S(R ) ,→ (B, k · kB ) ,→ S (R )

d is called a TF-homogeneous Banach space if S(R ) is dense in (B, k · kB ) and TF-shifts act isometrically on (B, k · kB ), i.e. if

d d kπ(λ)f kB = kf kB , ∀λ ∈ R × Rb , f ∈ B. (18) For such spaces the mapping λ → π(λ)f is continuous from d d R × Rb to (B, k · kB ). If it is not continuous on often has the adjoint action on the dual space of such TF-homogeneous L∞ d
M d Banach spaces (e.g. (R ), k · k∞ or ( b(R ), k · kMb )).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem TF-homogeneous Banach Spaces II

An important fact concerning this family is the minimality property S d
of the Segal algebra 0(R ), k · kS0 . Theorem There is a smallest member in the family of all TF-homogeneous Banach spaces, namely the Segal algebra S d
W L1 1 d 0(R ), k · kS0 = (F , ` )(R ). There is also a maximal space in the family of Fourier standard 0 d S 0 spaces, namely the dual space ( 0 (R ), k · kS0 ) resp. ∞ ∞ d W (FL , ` )(R ). The second claim even makes sense if FouSSs are defined as 0 d subspaces of the much larger space S (R ) of tempered distributions!

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Discussion of the Diagram

For each of the Fourier Standard Spaces the discussion of the above diagram makes sense. One may see that it can collaps totally to a single space, or that it has in fact a rich (like d
C0(R ), k · k∞ ) or simple structure. Theorem A Fourier standard space is maximal, i.e. coicides with AG GA Be = B = B if and only if B is a dual space. 0 There is also a formula for the predual spaces, it is ((B0) )0, where d d B0 = BAG = BGA is just the closure of S(R ) resp. S0(R ) in B.

d Of course (B, k · kB ) is minimal if and only if S0(R ) is a dense subspace of (B, k · kB ), resp. if it is a TF-homogeneous Banach space.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Discussion of the Diagram II

Theorem A Fourier standard space is reflexive if and only if both the space (B, k · kB ) and its dual are both minimal and maximal. In other words, for the space itself and its dual the diagram is reduced to a single space (B, k · kB ).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Constructions within the FouSS Family

1 taking Fourier transforms; 2 conditional dual spaces, i.e. the dual space of the closure of S0(G) within (B, k · kB ); 1 2 3 with two spaces B , B : take intersection or sum q 1 1 4 forming amalgam spaces W (B, ` ); e.g. W (FL , ` ); 5 defining pointwise or convolution multipliers; 6 using complex (or real) interpolation methods, so that we get the (Fourier invariant) spaces Mp,p = W (FLp, `p); 7 fractional invariant kernel and hull: For any given standard space (B, k · kB ) we could define the largest Banach space inside of B which is invariant under all the fractional FTs, or the smallest such space which allows a continuous embedding of (B, k · kB ) into that space.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Constructions within the FouSS Family II

To explain the setting let us start with the familiar family of p d L -spaces on a LCA group, say G = R , and p d
(B, k · kB ) = L (R ), k · kp , for some p ∈ [1, ∞). p d
The FL (R ), k · kp is well defined as the image of p d
L (R ), k · kp under the Fourier transform, with transport of norm. It is another FouSS, even for p > 2 (because it is still well 0 d S 0 defined as a subspace of ( 0 (R ), k · kS0 )). It is a natural question to find the range of values (r, s) such that

W (FLp, `r ) ⊆ FLp ⊆ W (FLp, `s ).

Investigations by Peter Gr¨obnerhave shown (1992) that this is OK if and only iff r ≤ min(p, p0) and s ≥ max(p, p0).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Constructions within the FouSS Family III

Modulation spaces are Fourier Standard Spaces p,q d
The unweighted modulation spaces M (R ), k · kMp,q can be obtained by first forming the Wiener amalgams W (FLp, `q) and then taking the inverse Fourier transform of these spaces. The above inclusion relations then translate into exact embedding conditions between Lp-spaces and the corresponding modulation spaces. Obviously there are natural embeddings between modulation spaces with parameters p1, q1 and p2, q2, with d 1 1,1 S0(R ) = M = M being the smallest one!

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Constructions within the FouSS Family IV

There is a small body of literature (mostly papers by Kelly McKennon, a former PhD student of Edwin Hewitt) concerning spaces of “tempered elements”. He has done the case starting B = Lp(G), over general LC groups, but the construction makes sense if (and only if) one has a nice invariant space which happens not to be a convolution (or pointwise) algebra. By intersecting the space with its own “multiplier algebra” one obtains an (abstract) Banach algebra, and often the Banach algebra homomorphism of this new algebra “are” just the translation invariant operators on the original spaces. p d
For the case of B = L (R ), k · kp one would define

Lt Lp Lp Lp p := ∩ HG ( , ).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Tempered elements in Lp-spaces

understood as the intersection of two FouSSs, with the natural norm, which is the sum of the Lp-norm of f plus the operator norm of the convolution operator. For p > 2 one has to be careful and has to define that operator d norm only by looking at the action of k → k ∗ f on Cc (R )! (convolution in the pointwise sense might fail to exist, on more than just a null-set!). However it is not a problem to approximate every element (in norm ∗ d or even just in the w -sense by test-functions in S0(R ) and then take the limit of the convolution products of the regularized expressions.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Constructions within the FouSS Family V

Another interesting result that came recently to my attention (thanks to Werner Ricker) provides an answer to the following question related to the Theorem of Hausdorff-Young: p d q d We know, that one has that FL (R ) ⊂ L (R ) for 1/p + 1/q = 1, whenever p ∈ [1, 2]. But is there any strictly larger, solid Banach space (B, k · kB ) (meaning pointwise ∞ d q d L (R )-module) such that it is still true that F(B) ⊂ L (R )). The answer can be descibed as the FouSS (with natural norm):

∞ p B = HL∞ (L , FL ).

∞ d p d In words: the pointwise multipliers from L (R ) to FL (R ).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Tensor products

Let us recall some basic terms concerning tensor products of functions or distributions (see [1], [2]) 1 2 d 1 2 Given two functions f and f on R respectively, we set f ⊗ f

1 2 1 2 d f ⊗ f (x1, x2) = f (x1)f (x2), xi ∈ R , i = 1, 2.

For distributions this definition can be extended by taking ∗ w -limits or by duality, just like µ1 ⊗ µ2 is defined, for two d bounded measures µ1, µ2 ∈ Mb(R ). 0 2d It is important to know that we have σ1 ⊗ σ2 ∈ S0 (R ) for any 0 d pair of distributions σ1, σ2 ∈ S0 (R ). S 0 d S 0 d In particular 0 (R )⊗b 0 (R ) is well defined and a (proper) 0 d subspace of S0 (R ).

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Tensor product of FouSSps

1 2 0 d Given two Banach spaces B and B embedded into S (R ), B1⊗ˆ B2 denotes their projective tensor product, i.e.

n X 1 2 X 1 2 o f | f = fn ⊗ fn , kfn kB1 kfn kB2 < ∞ ; (19)

It is easy to show that this defines a Banach space of tempered 2d distributions on R with respect to the (quotient) norm:

X 1 2 kf k⊗ˆ := inf { kfn kB1 kfn kB2 , ..} (20) where the infimum is taken over all admissible representations.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem 2d The Varopoulos algebra V0(R ) and bimeasures

For questions of harmonic analysis the so-called Varopoulos algebra V 2d C d C d 0(R ) := 0(R )⊗b 0(R ) plays an important role. The dual space of this tensor product, which is a proper subspace 2d 2d of C0(R ) is called the space of bi-measures BM(R ), which form again a Banach algebra with respect to convolution. 0 Their Fourier transforms (in the sense of S0 ) are still well defined, and are bounded continuous functions, and one has again a convolution theorem (convolution goes into pointwise multiplication under the FT). d The space BM shares with Mb(R ) the property that the compactly supported elements are dense in the space, i.e. B = BA in the spirit of the diagram.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The KERNEL THEOREM for ScRd

The kernel theorem for the Schwartz space can be read as follows: Theorem d 0 d For every continuous linear mapping T from S(R ) into S (R ) 0 2d there exists a unique tempered distribution σ ∈ S (R ) such that

d T (f )(g) = σ(f ⊗ g), f , g ∈ S(R ). (21)

0 2d Conversely, any such σ ∈ S (R ) induces a (unique) operator T such that (21) holds.

d The proof of this theorem is based on the fact that S(R ) is a nuclear Frechet space, i.e. has the topology generated by a sequence of semi-norms, can be described by a metric which d turns S(R ) into a complete metric space.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The KERNEL THEOREM for S0 I

Tensor products are also most suitable in order to describe the set of all operators with certain mapping properties. The backbone of the corresponding theorems are the kernel-theorem which reads as S d
follows (!! despite the fact that 0(R ), k · kS0 is NOT a nuclear Frechet space) One of the corner stones for the kernel theorem is: One of the d most important properties of S0(R ) (leading to a characterization given by V. Losert, [6]) is the tensor-product factorization: Lemma

k n ∼ k+n S0(R )⊗ˆ S0(R ) = S0(R ), (22) with equivalence of the corresponding norms.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The KERNEL THEOREM for S0 II

0 The Kernel Theorem for general operators in L(S0, S0 ): Theorem d 0 d If K is a bounded operator from S0(R ) to S0 (R ), then there 0 2d exists a unique kernel k ∈ S0 (R ) such that hKf , gi = hk, g ⊗ f i d for f , g ∈ S0(R ), where g ⊗ f (x, y) = g(x)f (y). Formally sometimes one writes by “abuse of language” Z Kf (x) = k(x, y)f (y)dy Rd with the understanding that one can define the action of the 0 d functional Kf ∈ S0 (R ) as Z Z Z Z Kf (g) = k(x, y)f (y)dy g(x)dx = k(x, y)g(x)f (y)dxdy. Rd Rd Rd Rd

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The KERNEL THEOREM for S0 III

This result is the “outer shell” of the Gelfand triple isomorphism. The “middle = Hilbert” shell which corresponds to the well-known 2 d result that Hilbert Schmidt operators on L (R ) are just those compact operators which arise as integral operators with 2 2d L (R )-kernels.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The KERNEL THEOREM for S0 IV

Theorem The classical kernel theorem for Hilbert Schmidt operators is 0 unitary at the Hilbert spaces level, with hT , SiHS = trace(T ∗ S ) as scalar product on HS and the usual Hilbert space structure on 2 2d L (R ) on the kernels. 2d Moreover, such an operator has a kernel in S0(R ) if and only if 0 d d the corresponding operator K maps S0 (R ) into S0(R ), but not only in a bounded way, but also continuously from w ∗−topology d into the norm topology of S0(R ). In analogy to the matrix case, where the entries of the matrix

ak,,j = T (ej )k = hT (ej ), ek i

we have for K ∈ S0 the continuous version of this principle: d K(x, y) = δx (T (δy ), x, y ∈ R .

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The Kernel Theorem as a BGT isomorphism

The different version of the kernel theorem for operators between 0 S0 and S0 can be summarized using the terminology of Banach Gelfand Triples (BGTR) as follows. Theorem There is a unique Banach Gelfand Triple isomorphism between the 2 0 2d Banach Gelfand triple of kernels (S0, L , S0 )(R ) and the operator Gelfand triple around the Hilbert space HS of Hilbert 0 0 Schmidt operators, namely (L(S0 , S0), HS, L(S0, S0 )), where the first set is understood as the w ∗ to norm continuous operators 0 d d from S0 (R ) to S0(R ), the so-called regularizing operators.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Fourier Standard Spaces of Operators

The kernel theorem allows to identify many spaces of linear operators (with different forms of continuity) with suitable FouSSs 2d over R . For example, there are the so-called Schatten classes of operators 2 d on the Hilbert space L (R ) which are compact operators with singular values in `p, for 1 ≤ p < ∞. These spaces are operator ideals within L(H), i.e. they are Banach spaces, continuously embedded into the space of compact operators over the Hilbert space H, as well as two-sided Banach ideals, i.e. whenever one has an operator T in such a space, and two bounded operators S1, S2 on H, then S1 ◦ T ◦ S2 also belongs to that operator ideal and the operator ideal norm is bounded by the operator ideal norm of T multiplied with the operator norms of S1 and S2.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Spaces of Operators

Another family of operators are defined by their boundedness between certain FouSSs, e.g. an operator may be bounded from p d q d L (R ) (with p ∈ [1, ∞)) to some L (R ), with 1 ≤ q ≤ ∞. Each of these operators has a kernel, so we can look at the set of p d q d all the kernels of bounded operators from L (R ) to L (R ) for p, q in the range described above, simply by testing the norm p d continuity on the dense subspace (of L (R ), for p < ∞) and 0 d S 0 embedding the target space into ( 0 (R ), k · kS0 ). Theorem Consider the Banach space of operators L(Lp, Lq), with 1 ≤ p, q < ∞, which is isomorphic to a space of kernels in 0 2d S0 (R ), with the norm of the kernel being just the operator norm of the corresponding operator. Then the space of kernels is ismorphic to the dual of the FouSS p d q0 d L (R )⊗bL (R ). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Spaces of Operators II

p d Next we define the Herz algebras A (R ) via the “convolution tensor product: The dual space of the space of all (convolution) p d p d multipliers from L (R ) to L (R ) (for 1 < p < ∞) can be d identified with the dual space of the Herz algebra Ap(R ), given by A d Lp d Lq d p(R ) := (R ) ∗b (R ). In the background of such a theorem stands the fact that a matrix commutes with (cyclic) translations if and only if it is constant along the side-diagonals. The kernels of such operators are constant along the main diagonal, respectively are a “moving average”. Spectral synthesis results for the Fourier 0 d transform on S0 (R ) then allow to derive this result.

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Spaces of Operators III

Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem References

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Elena Cordero, Hans G. Feichtinger, and Franz Luef. Banach Gelfand triples for Gabor analysis. In Pseudo-differential Operators, volume 1949 of Lecture Notes in Mathematics, pages 1–33. Springer, Berlin, 2008.

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Viktor Losert. A characterization of the minimal strongly character invariant Segal algebra. Ann. Inst. Fourier (Grenoble), 30:129–139, 1980.

Ferenc Weisz. Inversion of the short-time Fourier transform using Riemannian sums. J. Fourier Anal. Appl., 13(3):357–368, 2007. Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem