TIME-FREQUENCY ANALYSIS AND COORBIT SPACES OF OPERATORS MONIKA D ORFLER FRANZ LUEF HENRY MCNULTY

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TIME-FREQUENCY ANALYSIS AND COORBIT SPACES

OF OPERATORS

MONIKA D ¨

ORFLER, FRANZ LUEF, HENRY MCNULTY,

AND EIRIK SKRETTINGLAND

Abstract. We introduce an operator valued Short-Time Fourier Trans-

form for certain classes of operators with operator windows, and show

that the transform acts in an analogous way to the Short-Time Fourier

Transform for functions, in particular giving rise to a family of vector-

valued reproducing kernel Banach spaces, the so called coorbit spaces,

as spaces of operators. As a result of this structure the operators gen-

erating equivalent norms on the function modulation spaces are fully

classiﬁed. We show that these operator spaces have the same atomic

decomposition properties as the function spaces, and use this to give a

characterisation of the spaces using localisation operators.

1. Introduction

In time-frequency analysis, the modulation spaces Mp,q

mpRdq, ﬁrst intro-

duced by Feichtinger in 1983 [16], play a central role, where they deﬁne

spaces of functions with certain desirable time-frequency decay. In particu-

lar the Feichtinger algebra, M1pRdqor S0pRdq[14] [13], gives well concen-

trated functions in the time-frequency sense, which are for many purposes

the ideal atoms for Gabor analysis. The modulation spaces are usually de-

ﬁned in terms of the Short-Time Fourier Transform (STFT), namely as the

spaces

Mp,q

mpRdq:“ tψPS1pRdq:´żRd´żRd

|Vφ0ψpzq|pmpx, ωqpdx¯q{pdω¯1{qă 8u,

where φ0is the Gaussian. Modulation spaces and their various generali-

sations have been studied extensively, and surveys and monographs can be

found in [25] [6] [22]. The properties and utility of these function spaces are

too broad to hope to cover, but of particular interest to our work is that

these spaces are the coorbit spaces [17] [18] of the projective unitary repre-

sentation of the reduced Weyl-Heisenberg group, and as such have (among

others) the following properties:

(1) All gPL2pRdqthat satisfy the condition VggPL1

vpR2dq, generate

the same modulation spaces Mp,q

mpRdqas windows, and their norms

are equivalent.

(2) (Correspondence Principle) Given an atom gas above, there is an

isometric isomorphism Mp,q

mpRdq – tFPLp,q

mpR2dq:F“F6Vggu

2020 Mathematics Subject Classiﬁcation. 40E05; 47G30; 47B35; 47B10.

Key words and phrases. Operator-valued short-time Fourier transform, vector-valued

reproducing kernel Hilbert spaces, coorbit spaces of operators, Toeplitz operators.

arXiv:2210.04844v2 [math.FA] 7 Jun 2023

2MONIKA D ¨

ORFLER, FRANZ LUEF, HENRY MCNULTY, AND EIRIK SKRETTINGLAND

(where 6is the twisted convolution discussed below), given by Vg.

Note that the later are reproducing kernel Banach spaces.

There is a vast body of contributions to the theory of coorbit spaces, e.g.

[4, 8, 24]. In this work we examine spaces of operators exhibiting similar

properties, by introducing an STFT with operator window and argument, re-

turning an operator-valued function on phase space. One motivation comes

from [12], where local structures of a data set D“ tf1, ..., fNuwere identi-

ﬁed via mapping the data points of functions fion Rdto rank-one operators

fibfi, and constructing the data operator SD“řN

i“1fibfi. Hence, it

would be of interest to compare to data sets Dand D1via its respective

data operators SDand SD1. Another source of inspiration is the work [26],

where operator analogues of the Schwartz class of functions and of the space

of tempered distributions have been introduced and their basic theory has

been developed along the lines of the function/distribution case.

The concept of an STFT for operators is not a new one. In [5], the

authors consider the wavelet transform for the representation πpwqbπpzqon

HS “L2pRdq b L2pRdqto examine kernel theorems for coorbit spaces. This

entails using the standard scalar-valued construction for the coorbit spaces

deﬁned by the wavelets transform, giving diﬀerent spaces to our approach.

On the other hand nor are vector-valued reproducing kernel Hilbert spaces in

time-frequency analysis a new concept. In [2] and [1] an STFT is constructed

for vectors of functions, which results in a direct sum of Gabor spaces. Our

work diﬀers from these in that windows, arguments and resulting output of

the operator STFT are all operators.

In [34], the author introduced an equivalent notion of a STFT with an

operator window, given by

VSψpzq:“Sπpzq˚ψ(1)

for some appropriate operator Sand function ψ. In particular, the ques-

tion was considered of which operators would deﬁne equivalent norms on

Mp,q

mpRdqunder this STFT, that is, for which operators

}ψ}Mp,q

m— }Sπpzq˚ψ}Lp,q

mpR2d;L2q.

In further work by Guo and Zhao [23], some equivalent conditions for equiva-

lence were given. In both works a class of operators with adjoints in a certain

class of nuclear operators was discussed, along with the open question in the

latter of whether these operators exhausted all possible operators generating

equivalent norms on Mp,q

mpRdq. In this work we present an extension of the

operator window STFT (1), which acts on operators instead of functions.

We initially deﬁne such a transform for S, T PHS in the following manner:

Deﬁnition 1.1. (Operator STFT) For S, T PHS, the STFT of Twith

window S, is given by

VSTpzq:“S˚πpzq˚T.(2)

Note that in the case of rank-one operators S“gbeand T“fbefor

e, f, g PL2pRdqthe operator STFT becomes Vgfpzqebe, which is the STFT

of functions embedded into the space of Hilbert-Schmidt operator-valued

functions.

TIME-FREQUENCY ANALYSIS AND COORBIT SPACES OF OPERATORS 3

We examine the behaviour of this transform, e.g. Moyal’s identity, paying

particular attention to the spaces it produces as images. In this respect the

ﬁrst result of this paper demonstrates a parallel to the STFT of functions,

regarding the reproducing structure of the image of the Hilbert space of

Hilbert-Schmidt operators:

Theorem 1.2. For any Hilbert-Schmidt operator S, the space deﬁned by

VSpHSq:“ tVSTpzq:TPHSu

is a vector-valued uniform reproducing kernel Hilbert space as a subspace of

the Bochner-Lebesgue space L2pR2d;HSq.

Motivated by this, we extend the reproducing properties of this space

to the ”coorbit spaces”, and consider the spaces Av:“ tSPHS :VSSP

vpR2d;HSqu, and Mp,q

m:“ tTPS1:VSTPLp,q

mpR2d;HSqu, where S1are

operators with Weyl symbols in S1and SPAv, to derive the result

Theorem 1.3. For any SPM1

v, we have an isometric isomorphism

Mp,q

m– tΨPLp,q

mpR2d;HSq: Ψ “Ψ6VSSu

under the mapping

TÞÑ VST,

the twisted convolution 6is to be deﬁned in Section 2.3. Furthermore,

for all SPAvthe resulting spaces coincide, and the associated norms are

equivalent. The dual space of Mp,q

mis Mp1,q1

1{m, where 1

p`1

p1“1, 1

q`1

q1“1

with the usual adjustment for p, q “1,8. As a corollary of the coorbit

structure and independence of windows, we characterise operators satisfying

the equivalent norm condition;

Corollary 1.4. The operators which deﬁne equivalent norms on the spaces

Mp,q

mpRdqby

}S˚πpzq˚ψ}Lp,q

mpR2d;L2pRdqq

or every 1ďp, q ď 8 and v-multiplicative m, are precisely the admissible

operators

Av:“ tS:VSSPL1

vpR2d;HSqu,

We ﬁnally consider the atomic decomposition of operators in the Mp,q

which follows from the same arguments as the function case given the coor-

bit structure. Using this we can characterise the spaces using localisation

operators:

Corollary 1.5. Let φPL2pRdqbe non-zero and hPL1

vpR2dqbe some non-

negative symbol satisfying

Aďÿ

λPΛ

hpz´λq ď B

for positive constants A, B, and almost all zPR2d. Then for every v-

moderate weight mand 1ďpă 8 the operator TPM8

1{vbelongs to Mp,q

mif

and only if

␣Aφ

hπpλq˚T(λPΛPlp,q

mpΛ; HSq.

4MONIKA D ¨

ORFLER, FRANZ LUEF, HENRY MCNULTY, AND EIRIK SKRETTINGLAND

where Λ“αZˆβZis some full rank lattice.

2. Preliminaries

2.1. Time-Frequency Analysis Basics. While coorbit spaces are deﬁned

in general for integrable representations of locally compact groups, modu-

lation spaces of functions and the spaces discussed in this work arise from

the particular case of the time-frequency shifts πpzq, the projective unitary

representation of the reduced Weyl-Heisenberg group on the Hilbert space

L2pRdq. Such shifts can be deﬁned as the composition of the translation

operator Tx:fptq ÞÑ fpt´xq, and the modulation operator Mω:fptq ÞÑ

e2πiωtfptq, by the identity

πpzq “ MωTx

where z“ px, ωq P R2d. Direct calculations show that πpzqis unitary on

L2pRdq, and that we have

πpzqπpz1q “ e´2πiω1xπpz`z1q

πpzq˚“e´2πixωπp´zq.

The Short-Time Fourier Transform (STFT) for functions is then deﬁned, for

two functions f, g PL2pRdq, by

Vgfpzq:“ xf, πpzqgyL2.(3)

The window function gis usually chosen to have compact support, or be

concentrated around the origin, such as in the case of the normalised Gauss-

ian φ0ptq “ 2d{4eπt2. For f, g PL2pR2dq,Vgfis uniformly continuous as a

function in L2pR2dq, which will be instructive when considering reproducing

kernel Hilbert spaces later. One has for the STFT Moyal’s Identity (see

for example Theorem 3.2.1 of [22]), giving an understanding of the basic

properties of the STFT in terms of its window:

Lemma 2.1. (Moyal’s Identity) Given functions f1, f2, g1, g2PL2pRdq, we

have Vg1f1, Vg2f2PL2pR2dq, and in addition:

xVg1f1, Vg2f2yL2pR2dq“ xf1, f2yL2pRdqxg1, g2yL2pRdq.

As a direct consequence, we have that for any gPL2pRdqsuch that

}g}L2“1, the map Vg:L2pRdq Ñ L2pR2dqis an isometry. As such, we

can consider the inverse mapping. Rearranging Moyal’s identity shows the

reconstruction formula

f“żR2d

Vgfpzqπpzqg dz,(4)

for any gPL2pRdqwith }g} “ 1. A direct calculation then shows that the

adjoint V˚

gis given by

V˚

gpFq:“żR2d

Fpzqπpzqg dz,(5)

where the integral can be interpreted in the weak sense, and so from the

reconstruction formula

V˚

gVg“IL2pRdq.

TIME-FREQUENCY ANALYSIS AND COORBIT SPACES OF OPERATORS 5

2.2. Weight functions and mixed-norm spaces. We begin by deﬁning

a sub-multiplicative weight vas a non-negative, locally integrable function

on phase space R2dsatisfying the condition

vpz1`z2q ď vpz1qvpz2q

for all z1, z2PR2d. As a direct result, vp0q ě 1. A v-moderate weight mis

then a non-negative, locally integrable function on phase space such that

mpz1`z2q ď vpz1qmpz2q

for all z1, z2PR2d. As a particular consequence, we have for such a v, m

that

Cv,mvpzqďmpzq ď Cv,mvpzq.

In this work we consider weights of at most polynomial growth. We deﬁne

the weighted, mixed-norm space Lp,q

mpR2dq, for 1 ďp, q ă 8, as the functions

for which the norm

}F}Lp,q

m:“´żRd´żRd

|Fpx, ωq|pmpx, ωqpdx¯q{pdω¯1{q

is ﬁnite. In the case where por qis inﬁnite, we replace the correspond-

ing integral with essential supremum. For such spaces we have the duality

pLp,q

mpR2dqq1“Lp1,q1

1{mpR2dq, where 1

p`1

p1“1, 1

q`1

q1“1. Further details

on weights and mixed-norm spaces can be found in chapter 11, [22]. In this

work we consider discretisation over the full rank lattice Λ “αZdˆβZd.

An arbitrary lattice Λ “AZ2d,APGLp2d;Rqcan also be used, but the no-

tion of mixed-norm becomes less clear. We deﬁne the mixed-norm weighted

sequence space lp,q

mpΛ; HSqas the sequences apk,lqsuch that

}a}lp,q

mpΛ;HSq:“´ÿ

nPZd`ÿ

kPZd

mpαk, βlqp}aαk,βl}p

HS ˘q{p¯1{qă 8.

The Wiener Amalgam spaces introduced in [15] provide the required frame-

work for sampling estimates on the lattice. To that end we deﬁne for a given

function Ψ : R2dÑHS the sequence

aΨ

pk,lq“´ess sup

x,ωPr0,1sd

}Ψpx`k, ω `lq}HS ¯pk,lq.

Deﬁnition 2.2. Let 1ďp, q ď 8 and mbe some weight function. The

Wiener Amalgam space WpLp,g

mpR2d;HSqq consists of all functions Ψ : R2dÑ

HS such that

}aΨ

pk,lq}lp,q

mă 8,

with the norm }Ψ}WpLp,q

mpR2d;HSqq :“ }aΨ

pk,lq}lp,q

One feature of the Wiener Amalgam spaces we use (see for example Propo-

sition 11.1.4 of [22]) is the following:

Proposition 2.3. Let Λ“αZdˆβZdand ΨPWpLp,q

mpR2d;HSqq be con-

tinuous. Then

}Ψ|Λ}lp,q

˜mpΛ;HSqďc}Ψ}WpLp,q

mpR2d;HSqq

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TIME-FREQUENCYANALYSISANDCOORBITSPACESOFOPERATORSMONIKAD¨ORFLER,FRANZLUEF,HENRYMCNULTY,ANDEIRIKSKRETTINGLANDAbstract.WeintroduceanoperatorvaluedShort-TimeFourierTrans-formforcertainclassesofoperatorswithoperatorwindows,andshowthatthetransformactsinananalogouswaytotheShort-TimeFourierTransformforfunc...

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