A BANACH GELFAND TRIPLE motivated by and useful for time ...

Numerical Harmonic Analysis Group

A BANACH GELFAND TRIPLEmotivated by and useful for time-Frequency

An easy path to distribution theoryalso suitable for engineers

Hans G. [email protected]

www.nuhag.eu

Eotvos University H-1117 Budapest: July 4th, 2011

Hans G. Feichtinger [email protected] www.nuhag.euA BANACH GELFAND TRIPLE motivated by and useful for time-Frequency An easy path to distribution theory also suitable for engineers

OVERVIEW

The GOAL of this presentation is to convey theconcepts of Gelfand Triples, in particularBanach Gelfand Tripels, but also Banach frames bydescribing them and show their usefulness in thecontext of mathematical analysis, in particulartime-frequency analysis

Recall some concepts from linear algebra, especially that of agenerating system, a linear independent set of vectors, andthat of the dual vector space;

already in the context of Hilbert spaces the question arises:what is a correct generalization of these concepts?

Banach Gelfand Triple (comparable to rigged Hilbertspaces) are one way out;

Hans G. Feichtinger A BANACH GELFAND TRIPLE motivated by and useful for time-Frequency An easy path to distribution theory also suitable for engineers

A suitable Banach space of test functions & distributions

S0Schw L1

Tempered Distr.

SO’

L2

C0

FL1


... there is an implicit message:

Aside from the various technical terms coming up I hope to conveyimplicitly a few other messages, closely related to my view ofHarmonic Analysis as the sub-field of functional analysis which hasto do with group actions:

staying with Banach spaces and their duals one can doamazing things (avoiding topological vector spaces, Lebesgueintegration, or Schwartz distribution theory);

alongside with the norm topology just the very naturalw∗-topology, just in the form of pointwise convergence offunctionals, for the dual space has to be kept in mind(allowing the use of non-reflexive Banach spaces);

diagrams and operator descriptions allow to naturallygeneralize concepts from finite dimensional theory up tothe category of Banach Gelfand triples.


A picture of the singing of Pavarotti


compared to musical score ...


The analysis of a synthetic sound example


The two main difficulties in Gabor Analysis

There has been a very important influence on the understanding ofsignal expansions by the work of Denis Gabor (1946) which isrelated to the discussion of bases in vector spaces of signals.These difficulties are related to two short-comings which arise (infact very naturally) in the context of Gabor analysis, if one has the“trivial” linear algebra situation in mind:

1 Given a finite-dimensional vector space it makes sense tomostly work with bases. They are either maximal (finite)sequence of vectors (by enlarging linear independent vectors),or by searching for minimal generating sets (obtained byremoving elements from any given generating set);

2 While one has in the setting of signals of finite length boththe basis of unit vectors resp. the orthogonal basis of purefrequencies characters are not anymore in the Hilbertspace L2(R) of signals(of finite energy).


Denis Gabor’s suggestion II

We will discuss Gabor’s suggestion of 1946 in more detail, butbasically his reasoning was based on the following (good andinteresting) idea: Using the Gauss-function as a building block,which is optimally concentrated in a TF-sense (it provides equalityin the Heisenberg uncertainty) he suggested to represent arbitrarysignals/functions as superpositions of TF-shifted version of theGauss-functions along some lattice of the form aZ× bZ, witha = b = 1. His reasoning was based on the following intuition:

If a · b > 1 then one does not have sufficiently many buildingblocks in the Hilbert spaces in order to even approximategeneral functions in

(L2(R), ‖ · ‖2

). In fact, for any

g ∈ L2(R) (not just the Gauss function) the closed linear spanof such a Gabor family is a proper subspace of L2(R).If on the other hand a · b ¡ 1 then the corresponding familygets linear dependent, and therefore the representationwould not be unique anymore.


Denis Gabor’s suggestion III

As a consequence of these two observations he came to theconclusion that the choice a = 1 = b should be ideal, i.e.should allow for a unique representation of generalL2(R)-functions f , and thus give the coefficients a very clearmeaning.

While the suggestion was ignored by most mathematicians(except for A.J.E.M. Janssen, who tried to give it a precise,mathematical meaning using distribution theory) until the late70-th, the idea was developed further in the engineeringcommunity, typically with the comment that Gabor expansionsare “unfortunatly” numerically instable, although quiteintuitive (microtonal piano);


Denis Gabor’s suggestion IV

Let me give an interpretation of the idea, and formulate (thewishes, first) in a strict mathematical language.

It is obvious that the uniqueness of coefficients implies that(as in the linear algebra situation) the coefficient mappingassigning each f ∈ L2(R) its coefficient in such a systemwould be linear. It is quite sure, that D. Gabor would haveagreed (i.e. confirmed potential hopes) that those coefficientsshould be in `2(Z2) and that the coefficient mapping is alsocontinuous. In fact, if small changes in the signal wouldintroduce huge changes in the coefficients their usefulnesswould be very much spoiled. So in other words it is probablynot overinterpreting D. Gabor if one says that most likely hewas claiming that the suggested family should be aRiesz-basis for the Hilbert space

(L2(R), ‖ · ‖2

).


Denis Gabor’s suggestion V

One might argue that the price to be paid for the specific andinteresting structure of the Gabor family would by just thefact that one is loosing orthogonality (obviously: two shiftedGaussians are never perpendicular to each other in L2(R)!),but otherwise there should be some invertible (Banach-space)operator on L2(R) which allows to turn the Gabor family intoan orthonormal basis. In fact, nowadays we would argue thatone just has to the so called Lowdin orthogonalization, i.e.apply the inverse square root of the Gram-matrix of thissystem in order to obtain an orthonormal Riesz basis. If thiswas in fact possible one could even claim: replace theGauss-function by some other (Schwartz) function in L2(R),then you can even have an orthonormal basis of Gabor-type.

one can do a MATLAB experiment, with interesting results.


Finite Gabor families at critical density

Either the biorthogonal family is not well concentrated (and in factone dimension is lost!), or ...

−50 0 500

0.05

0.1

0.15

0.2

0.25

0.3

discrete Gauss atom, n = 144; a=12; b=12

−50 0 500

0.05

0.1

0.15

0.2

0.25

0.3

Fourier Gauss atom, n = 144; a=12; b=12

−50 0 50

−0.2

−0.1

0

0.1

0.2

0.3

0.4

dual Gauss atom, n = 144; a=12; b=12

−50 0 50

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fourier dual Gauss, n = 144; a=12; b=12



same for the (quasi-orthonormal) version (Lowdin). OR one takesthe “symmetric Gauss function” (invariant under the flip-operationin MATLAB): orthogonalization then:

−50 0 50

−0.05

0

0.05

0.1

0.15

0.2

0.25

orth. Gauss atom, n = 144; a=12; b=12

−50 0 50

−0.05

0

0.05

0.1

0.15

0.2

0.25

Fourier orth. Gauss atom, n = 144; a=12; b=12

−50 0 50−0.1

0

0.1

0.2

0.3alt. orth. Gauss atom, n = 144; a=12; b=12

−50 0 50

−0.05

0

0.05

0.1

0.15

0.2

0.25

alt. orth. dual Gauss, n = 144; a=12; b=12



The loss of one dimension (the orthogonal complement of therange is one-dimensional) is connected with the zero of theZak-transform of the Gauss function at (1/2, 1/2) (see left part ofthe picture);


What is known about Gabor families I

What we also know nowadays:

For any value of a, b the corresponding Gabor families arelinear independent in the classical sense (cf. Heil’sconjecture), i.e. finite linear combinations are zero if and onlyif the sequence of coefficients is trivial;

Gabor’s family is dense in L2(R), so finite linear combinationssuffice in order to approximate arbitrary L2(R)-functions.HOWEVER, if one wants to obtain a better approximation thecoefficients have to be recalculated from scratch, and thebetter the approximation, the higher the (cost, i.e. the)`2-norm of the corresponding coefficients will be;

the set is not even minimal in that respect, one can evenremove one element and still have the same property(however not two or more elements!)


What is known about Gabor families II

On the other hand we know positively:

when a = 1 = b then not every f ∈ L2(R) can be representedusing `2(Z2)-coefficients [not even bounded coefficientssuffice, according to A.J.E.M. Janssen];

If ab < 1 the corresponding Gabor family is a stablegenerating system (a so-called frame for L2(R));

any element in the system can be expressed as a linearcombination of the remaining ones, even when only`1-coefficients are allowed;

For ab < 1 the so-called dual frame (providing the minimal`2-coefficients) is in fact another Schwartz-function,hence the whole procedure has good locality.


Lessons to be learned from this situation

If one tries to imitate the situation envisaged by D. Gabor in the settingof the finite discrete group Z144 one comes to the followingconclusions/observations:

if one takes the correct Gauss-function (which is invariant under thediscrete Fourier transform) one is loosing in the critical case,a = 12 = b, one dimension, therefore one has to take - if theTF-structure is important - redundant families of TF-shifteddiscrete Gauss-functions;

Having taken this decision it is natural to obtain coefficients usingthe strategy of normal equations, i.e. to go for the MNLSQ-solutionof the problem. It turns out that this is possible (using theMoore-Penrose pseudo-inverse) and the resulting family isessentially another Gabor family, with the dual generator;

In a similar way, making the lattice constants larger onecan get fine linear independent Gabor families which havebiorthogonal families which are again of a Gaborian structure.


Choose Generating System or Lin. Independence

One possible conclusion from the above problems is the observationthat there might be some conflict with the wish of having a certainstructure of a (Riesz) basis, combined with good numericalstability as well as good localization in time and frequency.It turns out that one has to make a distinction, whether one wantsto expand arbitrary signals (then generating systems with a decentredundancy are preferred over those with minimal redundancy butbad condition number!), or whether (a stable form of) linearindependence is relevant: This is of course the case for(mobile/digital) communication.


Moving to the continuous setting, e.g. Rd

If one replaces the finite sequences with “function on the real lineR” or more general on Euclidian spaces Rn one faces extrachallenges, and questions like the following arise:

Is it true, that the synthesis mapping from `2(Z2d) intoL2(Rd) is continuous for every g ∈ L2(Rd)? (NO!);

Can one say that the (continuous) STFT (short-time Fouriertransform) of Vg (f ) ∈ L2(R2d) belongs to `2(Z2d)? (NO!);

If a Gabor family defines an orthonormal (or just Riesz) basisfor a given lattice, can we be sure that it is still a Riesz basisfor lattices which are closeby? (NO, take 1[0,1] and thestandard lattice Z2);

What about Gabor expansions for Lp-functions?


A Typical Musical STFT

A typical waterfall melody (Beethoven piano sonata) depicturedusing the spectrogram, displaying the energy distribution in the TF= time-frequency plan:


The key-players for time-frequency analysis

Time-shifts and Frequency shifts

Tx f (t) = f (t − x)

and x , ω, t ∈ Rd

Mωf (t) = e2πiω·t f (t) .

Behavior under Fourier transform

(Tx f ) = M−x f (Mωf ) = Tω f

The Short-Time Fourier Transform

Vg f (λ) = 〈f ,MωTtg〉 = 〈f , π(λ)g〉 = 〈f , gλ〉, λ = (t, ω);


The Schrodinger Representation

For people in representation theory I could explain the spectrogramis just displaying to you a typical representation coefficient of the(projective) Schrodinger Representation of the (reduced)Heisenberg Group Hd (for d = 1).According to Roger Howe this group has the phantastic “hinduisticmultiplicity in one” property of allowing a variety of differentlooking but in fact mathematically equivalent representations (dueto the von-Neumann uniqueness theorem), which indicates theconnection to quantum mechanics, the theory of coherent states,and related topics (where e.g. rigged Hilbert spaces, the bras andkets appear already), where concepts as described below are in factalso helpful (to put expressions such as continuous integralrepresentations on a firm mathematical ground); but we will startfrom known grounds...


Geometric interpretation of matrix multiplication

Null(A) ⊆ Rn

Row(A) Col(A) ⊆ Rm-

T = T|row(A)

inv(T )

?

PRow

@@@@@@@@@@R

T T ′

Rm ⊇ Null(A′)

?

�

PCol

��

��

��

?

T = T ◦ PRow , pinv(T ) = inv(T ) ◦ PCol .


Matrices of maximal rank

We will be mostly interested (as models for Banach Frames andRiesz projection bases) in the situation of matrices of maximalranks, i.e. in the situation where r = rank(A) = max(m, n), whereA = (a1, · · · , ak).Then either the synthesis mapping x 7→ A ∗ x =

∑k xkak has

trivial kernel (i.e. the column vectors of A are alinear independent set, spanning the column-space of which is ofdimension r = n), or the analysis mapping y 7→ A′ ∗ y = (〈y , ak〉)has trivial kernel, hence the column spaces equals the target space(or r = m), or the column vectors are a spanning set for Rm.


............... continued

For Riesz basic sequences we have the following diagram:

X

X0 Y-C�

R

P

?

@@@@@R

C

Definition

A sequence (hk) in a separable Hilbert space H is a Riesz basis forits closed linear span (sometimes also called a Riesz basicsequence) if for two constants 0 < D1 ≤ D2 <∞,

D1‖c‖2`2 ≤

∥∥∥∑k

ckhk

∥∥∥2

H≤ D2‖c‖2

`2 , ∀c ∈ `2 (1)

A detail description of the concept of Riesz basis can be found in([?]) where the more general concept of Riesz projection bases isexplained.


Reflect also for a moment about daily actions:

We are calculating with all kind of numbers in our daily life. Butjust recall the most beautiful equation

e2πi = 1.

It uses the exponential function, with a (purely) imaginaryexponent to get a nice result, more appealing than (the equivalent)

cos(2π) + i ∗ sin(2π) = 1 in C.

But actual computation are done for rational numbers only!! Recall

Q ⊂ R ⊂ C


Existing examples of Gelfand Triples

So-called Gelfand Triples are already widely used in various fields ofanalysis. The prototypical example in the theory of PDE iscertainly the Schwartz Gelfand triple, consisting of the space oftest functions S(Rd) of rapidly decreasing functions, denselysitting inside of

(L2(Rd), ‖ · ‖2

), which in turn is embedded into

the space of tempered distributions S ′(Rd).

S(Rd) ↪→ L2(Rd) ↪→ S ′(Rd). (2)

Alternatively (e.g. for elliptic PDE) one used

Hs(Rd) ↪→ L2(Rd) ↪→ H′s(Rd). (3)

It is obtained via the Fourier transform form

L2w (Rd) ↪→ L2(Rd) ↪→ L2

w (Rd)′. (4)


What is a generating set in a Hilbert space

We teach in our linear algebra courses that the following propertiesare equivalent for a set of vectors (fi )i∈I in V:

1 The only vector perpendicular to a set of vectors is ∅;2 Every v ∈ V is a linear combination of these vectors.

An attempt to transfer these ideas to the setting of Hilbert spacesone comes up with several different generalizations:

a family is total if its linear combinations are dense;

a family is a frame if there is a bounded linear mapping fromH into `2(I ) f 7→ c = c(f ) = (ci )i∈I such that

f =∑i∈I

ci fi ∀f ∈ H. (5)


The usual definition of frames

There is another, equivalent characterization of frames. First, it isan obvious consequence of the characterization given above, that

f =∑i∈I

ci fi ∀f ∈ H. (6)

implies that there exists C ,D > 0 such that

C‖f ‖2 ≤∑i∈I|〈f , fi 〉|2 ≤ D‖f ‖2 ∀f ∈ H. (7)

For the converse observe that Sf :=∑

i∈I 〈f , fi 〉fi is a strictly

positive definite operator and the dual frame (fi ) satisfies

f =∑i∈I〈f , fi 〉fi =

∑i∈I〈f , fi 〉fi


Frames and Riesz Bases: the Diagram

P = C ◦R is a projection in Y onto the range Y0 of C, thus wehave the following commutative diagram.

Y

X Y0-C

� R ?

P

��

��

R


[


Denis Gabor’s suggestion of 1946

There is one very interesting example (the prototypical problemgoing back to D. Gabor, 1946): Consider the family of alltime-frequency shifted copies of a standard Gauss functiong0(t) = e−π|t|

2(which is invariant under the Fourier transform),

and shifted along Z (Tnf (z) = f (z − n)) and shifted also in timealong Z (the modulation operator is given byMkh(z) = χk(z) · h(z), where χk(z) = e2πikz).Although D. Gabor gave some heuristic arguments suggesting toexpand every signal from L2(R) in a unique way into a (double)series of such “Gabor atoms”, a deeper mathematical analysisshows that we have the following problems (the basic analysishas been undertaken e.g. by A.J.E.M. Janssen in the early 80s):


TF-shifted Gaussians: Gabor families

−200 −100 0 100 200

0

0.1

0.2

the Gabor atom

−200 −100 0 100 200

0

2

4

6

FT of Gabor atom

−200 −100 0 100 200

0

0.1

0.2

time−shift of Gabor atom

−200 −100 0 100 200

−5

0

5

FT of time−shifted Gabor atom

−200 −100 0 100 200

−0.2

0

0.2

frequency−shifted Gabor atom

−200 −100 0 100 200

0

2

4

6

FT of frequ−shifted Gabor atom


Problems with the original suggestion

Even if one allows to replace the time shifts from along Z bytime-shifts along aZ and accordingly frequency shifts along bZ onefaces the following problems:

1 for a · b = 1 (in particular a = 1 = b) one finds a total subset,which is not a frame nor Riesz-basis for L2(R), which isredundant in the sense: after removing one element it is stilltotal in L2(R), while it is not total anymore after removal ofmore than one such element;

2 for a · b > 1 one does not have anymore totalness, but a Rieszbasic sequence for its closed linear span ( $ L2(R));

3 for a · b < 1 one finds that the corresponding Gaborfamily is a Gabor frame: it is a redundant familyallowing to expand f ∈ L2(R) using `2-coefficients (butone can remove infinitely many elements and still havethis property!);


Rethinking shortly the Fourier Transform

Since the Fourier transform is one of the central transforms, bothfor abstract harmonic analysis, engineering applications andpseudo-differential operators let us take a look at it first. People(and books) approach it in different ways and flavours:

It is defined as integral transform (Lebesgue!?);

It is computed using the FFT (what is the connection);

Should engineers learn about tempered distributions?

How can we reconcile mathematical rigor and still stay intouch with applied people (physics, engineering).


The finite Fourier transform (and FFT)

For practical applications the discrete (finite) Fourier transform isof upmost importance, because of its algebraic properties [jointdiagonalization of circulant matrices, hence fast multiplication ofpolynomials, etc.] and its computational efficiency(FFT algorithms of signals of length N run in Nlog(N) time, forN = 2k , due to recursive arguments).It maps a vector of length n onto the values of the polynomialgenerated by this set of coefficients, over the unit roots of order non the unit circle (hence it is a Vandermonde matrix). It is aunitary matrix (up to the factor 1/

√n) and maps pure frequencies

onto unit vectors (engineers talk of energy preservation).


The Fourier Integral and Inversion

If we define the Fourier transform for functions on Rd using anintegral transform, then it is useful to assume that f ∈ L1(Rd), i.e.that f belongs to the space of Lebesgues integrable functions.

f (ω) =

∫Rd

f (t) · e−2πiω·t dt (8)

The inverse Fourier transform then has the form

f (t) =

∫Rd

f (ω) · e2πit·ω dω, (9)

Strictly speaking this inversion formula only makes sense under theadditional hypothesis that f ∈ L1(Rd). One often speaks ofFourier analysis followed by Fourier inversion as a method tobuild f from the pure frequencies ( Fourier synthesis).


The classical situation with Fourier

Unfortunately the Fourier transform does not behave well withrespect to L1, and a lot of functional analysis went into fightingthe problems (or should we say symptoms?)

1 For f ∈ L1(Rd) we have f ∈ C0(Rd) (but not conversely, norcan we guarantee f ∈ L1(Rd));

2 The Fourier transform f on L1(Rd) ∩ L2(Rd) is isometric inthe L2-sense, but the Fourier integral cannot be writtenanymore;

3 Convolution and pointwise multiplication correspond to eachother, but sometimes the convolution may have to be taken asimproper integral, or using summability methods;

4 Lp-spaces have traditionally a high reputation amongfunction spaces, but tell us little about f .


A schematic description of the situation

L1

L2

C0

FL1

the classical Fourier situation


A schematic description (more details/spaces)

S0


The way out: Test Functions and Generalized Functions

The usual way out of this problem zone is to introduce generalizedfunctions. In order to do so one has to introduce test functions,and give them a reasonable topology (family of seminorms), sothat it makes sense to separate the continuous linear functionalsfrom the pathological ones. The “good ones” are admitted andcalled generalized functions, since most reasonable ordinaryfunctions can be identified (uniquely) with a generalized function(much as 5/7 is a complex number!).If one wants to have Fourier invariance of the space ofdistributions, one must Fourier invariance of the space of testfunctions (such as S(Rd)). If one wants to have - in addition -also closedness with respect to differentiation one has to take moreor less S(Rd). BUT THERE IS MORE!


A schematic description of the situation

S0Schw L1

Tempered Distr.

SO’

L2

C0

FL1


The Banach space(S0(Rd ), ‖ · ‖S0

)Without differentiability there is a minimal, Fourier andisometrically translation invariant Banach space (called(S0(Rd), ‖ · ‖S0

)or (M1(Rd), ‖ · ‖M1)), which will serve our

purpose. Its dual space (S0′(Rd), ‖ · ‖S0

′) is correspondingly thelargest among all Fourier invariant and isometrically translationinvariant “objects” (in fact so-called local pseudo-measures orquasimeasures, orginally introduced in order to describe translationinvariant systems as convolution operators).Although there is a rich zoo of Banach spaces around (one canchoose such a family, the so-called Shubin classes - to intersect inthe Schwartz class and their union is corresondingly S ′(Rd)), wewill restrict ourselves to the situation of Banach Gelfand Triples,mostly related to (S0,L

2,S0′)(Rd).


The S0-Banach Gelfand Triple

The S0 Gelfand triple

S0

S0’

L2


The key-players for time-frequency analysis

Time-shifts and Frequency shifts (II)

Tx f (t) = f (t − x)

and x , ω, t ∈ Rd

Mωf (t) = e2πiω·t f (t) .

Behavior under Fourier transform

(Tx f ) = M−x f (Mωf ) = Tω f

The Short-Time Fourier Transform

Vg f (λ) = 〈f ,MωTtg〉 = 〈f , π(λ)g〉 = 〈f , gλ〉, λ = (t, ω);


A Banach Space of Test Functions (Fei 1979)

A function in f ∈ L2(Rd) is in the subspace S0(Rd) if for somenon-zero g (called the “window”) in the Schwartz space S(Rd)

‖f ‖S0 := ‖Vg f ‖L1 =

∫∫Rd×Rd

|Vg f (x , ω)|dxdω <∞.

The space(S0(Rd), ‖ · ‖S0

)is a Banach space, for any fixed,

non-zero g ∈ S0(Rd)), and different windows g define the samespace and equivalent norms. Since S0(Rd) contains the Schwartzspace S(Rd), any Schwartz function is suitable, but alsocompactly supported functions having an integrable Fouriertransform (such as a trapezoidal or triangular function) aresuitable. It is convenient to use the Gaussian as a window.


Basic properties of M1 = S0(Rd )

Lemma

Let f ∈ S0(Rd), then the following holds:

(1) π(u, η)f ∈ S0(Rd) for (u, η) ∈ Rd × Rd , and‖π(u, η)f ‖S0 = ‖f ‖S0 .

(2) f ∈ S0(Rd), and ‖f ‖S0 = ‖f ‖S0 .

In fact,(S0(Rd), ‖ · ‖S0

)is the smallest non-trivial Banach space

with this property, and therefore contained in any of the Lp-spaces(and their Fourier images).


BANACH GELFAND TRIPLES: a new category

Definition

A triple, consisting of a Banach space B, which is dense in someHilbert space H, which in turn is contained in B′ is called aBanach Gelfand triple.

Definition

If (B1,H1,B′1) and (B2,H2,B

′2) are Gelfand triples then a linear

operator T is called a [unitary] Gelfand triple isomorphism if

1 A is an isomorphism between B1 and B2.

2 A is [a unitary operator resp.] an isomorphism between H1

and H2.

3 A extends to a weak∗ isomorphism as well as a norm-to-normcontinuous isomorphism between B′1 and B′2.


Banach Gelfand Triples, ctc.

In principle every CONB (= complete orthonormal basis)Ψ = (ψi )i∈I for a given Hilbert space H can be used to establishsuch a unitary isomorphism, by choosing as B the space ofelements within H which have an absolutely convergent expansion,i.e. satisfy

∑i∈I |〈x , ψi 〉| <∞.

For the case of the Fourier system as CONB for H = L2([0, 1]), i.e.the corresponding definition is already around since the times ofN. Wiener: A(U), the space of absolutely continuous Fourierseries. It is also not surprising in retrospect to see that the dualspace PM(U) = A(U)′ is space of pseudo-measures. One canextend the classical Fourier transform to this space, and in factinterpret this extended mapping, in conjunction with the classicalPlancherel theorem as the first unitary Banach Gelfand tripleisomorphism, between (A,L2,PM)(U) and (`1, `2, `∞)(Z).


The Fourier transform as BGT automorphism

The Fourier transform F on Rd has the following properties:

1 F is an isomorphism from S0(Rd) to S0(Rd),

2 F is a unitary map between L2(Rd) and L2(Rd),

3 F is a weak* (and norm-to-norm) continuous bijection fromS0′(Rd) onto S0

′(Rd).

Furthermore, we have that Parseval’s formula

〈f , g〉 = 〈f , g〉 (10)

is valid for (f , g) ∈ S0(Rd)× S0′(Rd), and therefore on each level

of the Gelfand triple (S0,L2,S0

′)(Rd).


FACTS

Grochenig and Leinert have shown (J. Amer. Math. Soc., 2004):

Theorem

Assume that for g ∈ S0(Rd) the Gabor frame operator

S : f 7→∑λ∈Λ

〈f , π(λ)g〉π(λ)g

is invertible as an operator on L2(Rd), then it is also invertible onS0(Rd) and in fact on S0

′(Rd).In other words: Invertibility at the level of the Hilbert spaceautomatically !! implies that S is (resp. extends to ) anisomorphism of the Gelfand triple automorphism for(S0,L

2,S0′)(Rd).


The w ∗− topology: a natural alternative

It is not difficult to show, that the norms of (S0,L2,S0

′)(Rd)correspond to norm convergence in (L1,L2,L∞)(R2d).The FOURIER transform, viewed as a BGT-automorphism isuniquely determined by the fact that it maps pure frequencies ontothe corresponding point measures δω.


Frames and Riesz Bases: the Diagram

P = C ◦R is a projection in Y onto the range Y0 of C, thus wehave the following commutative diagram.

Y

X Y0-C

� R ?

P

��

��

R


The frame diagram for Hilbert spaces:

`2(I )

H C(H)-C

� R ?

P

��

��

R


The frame diagram for Hilbert spaces (S0,L2,S0

′):

(`1, `2, `∞)

(S0,L2,S0

′) C((S0,L2,S0

′))-C

� R ?

P

��

��

R


Verbal Description of the Situation

Assume that g ∈ S0(Rd) is given and some lattice Λ. Then (g ,Λ)generates a Gabor frame for H = L2(Rd) if and only if thecoefficient mapping C from (S0,L

2,S0′)(Rd) into (`1, `2, `∞)(Λ) as

a left inverse R (i.e. R ◦ C = IdH ), which is also aGTR-homomorphism back from (`1, `2, `∞) to (S0,L

2,S0′).

In practice it means, that the dual Gabor atom g is also in S0(Rd),and also the canonical tight atom S−1/2, and therefore the wholeprocedure of taking coefficients, perhaps multiplying them withsome sequence (to obtain a Gabor multiplier) and resynthesis iswell defined and a BGT-morphism for any such pair.


Summability of sequences and quality of operators

One can however also fix the Gabor system, with both analysis andsynthesis window in S0(Rd) (typically one will take g and grespectively, or even more symmetrically a tight Gabor window).Then one can take the multiplier sequence in different sequencespaces, e.g. in (`1, `2, `∞)(Λ).

Lemma

Then the mapping from multiplier sequences to Gabor multipliersis a Banach Gelfand triple homomorphism into Banach Gelfandtriple of operator ideals, consisting of the Schatten classe S1 =trace class operators, H = HS, the Hilbert Schmidt operators, andthe class of all bounded operators (with the norm and strongoperator topology).


Automatic continuity (> Balian-Low)

In contrast to the pure Hilbert space case (the box-function is anideal orthonormal system on the real line, but does NOT allow forany deformation, without loosing the property of being even aRiesz basis):

Theorem (Fei/Kaiblinger, TAMS)

Assume that a pair (g ,Λ), with g ∈ S0(Rd) defines a Gabor frameor a Gabor Riesz basis respectively [note that by Wexler/Raz andRon/Shen these to situations are equivalent modulo taking adjointsubgroups!], then the same is true for slightly perturbed atoms orlattices, and the corresponding dual atoms (biorthogonalgenerators) depend continuously in the

(S0(Rd), ‖ · ‖S0

)-sense on

both parameters.


Invertibility, Surjectivity and Injectivity

In another, very recent paper, Charly Groechenig has discoveredthat there is another analogy to the finite dimensional case: Thereone has: A square matrix is invertible if and only if it is surjectiveor injective (the other property then follows automatically).We have a similar situation here (systematically describe inCharly’s paper):K.Grchenig: Gabor frames without inequalities, Int.

Math. Res. Not. IMRN, No.23, (2007).


Matrix-representation and kernels

We know also from linear algebra, that any linear mapping can beexpressed by a matrix (once two bases are fixed). We have asimilar situation through the so-called kernel theorem. It usesB = L(S0

′,S0).

Theorem

There is a natural BGT-isomorphism between (B,H,B′) and(S0,L

2,S0′)(R2d). This in turn is isomorphic via the spreading and

the Kohn-Nirenberg symbol to (S0,L2,S0

′)(Rd × Rd). Moreover,the spreading mapping is uniquely determined as theBGT-isomorphism, which established a correspondence betweenTF-shift operators π(λ) and the corresponding point masses δλ.


Kernel Theorem for general operators in L(S0,S0′)

Theorem

If K is a bounded operator from S0(Rd) to S0′(Rd), then there

exists a unique kernel k ∈ S0′(R2d) such that 〈Kf , g〉 = 〈k , g ⊗ f 〉

for f , g ∈ S0(Rd), where g ⊗ f (x , y) = g(x)f (y).

Formally sometimes one writes by “abuse of language”

Kf (x) =

∫Rd

k(x , y)f (y)dy

with the understanding that one can define the action of thefunctional Kf ∈ S0

′(Rd) as

Kf (g) =

∫Rd

∫Rd

k(x , y)f (y)dy g(x)dx =

∫Rd

∫Rd

k(x , y)g(x)f (y)dxdy .


Kernel Theorem II: Hilbert Schmidt Operators

This result is the “outer shell” of the Gelfand triple isomorphism.The “middle = Hilbert” shell which corresponds to the well-knownresult that Hilbert Schmidt operators on L2(Rd) are just thosecompact operators which arise as integral operators withL2(R2d)-kernels. The complete picture can be best expressed by aunitary Gelfand triple isomorphism. First the innermost shell:

Theorem

The classical kernel theorem for Hilbert Schmidt operators isunitary at the Hilbert spaces level, with 〈T , S〉HS = trace(T ∗ S ′)as scalar product on HS and the usual Hilbert space structure onL2(R2d) on the kernels. An operator T has a kernel inK ∈ S0(R2d) if and only if the T maps S0

′(Rd) into S0(Rd),boundedly, but continuously also from w∗−topology into the normtopology of S0(Rd).


Kernel Theorem III

Remark: Note that for such regularizing kernels in K ∈ S0(R2d)the usual identification. Recall that the entry of a matrix an,k isthe coordinate number n of the image of the n−th unit vectorunder that action of the matrix A = (an,k):

k(x , y) = T (δy )(x) = δx(T (δy )).

Note that δy ∈ S0′(Rd) implies that K (δy ) ∈ S0(Rd) by the

regularizing properties of K , hence the pointwise evaluation makessense.With this understanding our claim is that the kernel theoremprovides a (unitary) isomorphism between the Gelfand triple (ofkernels) (S0,L

2,S0′)(R2d) into the Gelfand triple of operator spaces

(L(S0′,S0),HS,L(S0,S0

′)).


AN IMPORTANT TECHNICAL warning!!

How should we realize these various BGT-mappings?Recall: How can we check numerically that e2πi = 1??Note: we can only do our computations (e.g. multiplication,division etc.) properly in the rational domain Q, we get to R byapproximation, and then to the complex numbers applying “thecorrect rules” (for pairs of real numbers).In the BGT context it means: All the (partial) Fourier transforms,integrals etc. only have to be meaningful at the S0-level!!! (noLebesgue even!), typically isometric in the L2-sense, and extend byduality considerations to S0

′ when necessary, using w∗-continuity!The Fourier transform is a good example (think of Fourierinversion and summability methods), similar arguments apply tothe transition from the integral kernel of a linear mapping to itsKohn-Nirenberg symbol., e.g..


The w ∗− topology: a natural alternative

It is not difficult to show, that the norms of (S0,L2,S0

′)(Rd)correspond to norm convergence in (L1,L2,L∞)(R2d).Therefore it is interesting to check what the w∗-convergence lookslike:

Lemma

For any g ∈ S0(Rd) a sequence σn is w∗-convergent to σ0 if andonly the spectrograms Vg (σn) converge uniformly over compactsets to the spectrogram Vg (σ0).

The FOURIER transform, viewed as a BGT-automorphism isuniquely determined by the fact that it maps pure frequencies ontothe corresponding point measures δω.


The w ∗− topology: dense subfamilies

From the practical point of view this means, that one has to lookat the spectrograms of the sequence σn and verify whether theylook closer and closer the spectrogram of the limit distributionVg (σ0) over compact sets.The approximation of elements from S0

′(Rd) takes place by abounded sequence.Since any Banach-Gelfand triple homomorphism preserves thisproperty (by definition) one can reduce many problems tow∗-dense subsets of

(S0(Rd), ‖ · ‖S0

).

Let us look at some concrete examples: Test-functions, finitediscrete measures µ =

∑i ciδti , trigonometric polynomials

q(t) =∑

i aie2πiωi t , or discrete AND periodic measures

(this class is invariant under the generalized Fourier transformand can be realized computationally using the FFT).


The w ∗− topology: approximation strategies

How to approximate general distributions by test functions:Regularization procedures via product convolution operators,hα(gβ ∗ σ)→ σ or TF-localization operators: multiply theSTFT with a 2D-summability kernel before resynthesis (e.g.partial sums for Hermite expansion);

how to approximate an L1-Fourier transform by test functions:and classical summability

how to approximate a test function by a finite disretesequence using quasi-interpolation (N. Kaiblinger):QΨf (x) =

∑i f (xi )ψi (x).


A BANACH GELFAND TRIPLE motivated by and useful for time ...

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