Quantum Harmonic Analysis on Lattices and Gabor Multipliers

https://doi.org/10.1007/s00041-020-09759-1

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Eirik Skrettingland¹

Received: 29 July 2019 / Revised: 7 April 2020 / Published online: 12 June 2020

© The Author(s) 2020

Abstract

We develop a theory of quantum harmonic analysis on lattices inR^2d. Convolutions of a sequence with an operator and of two operators are defined over a lattice, and using corresponding Fourier transforms of sequences and operators we develop a version of harmonic analysis for these objects. We prove analogues of results from classical harmonic analysis and the quantum harmonic analysis of Werner, including Taube- rian theorems and a Wiener division lemma. Gabor multipliers from time-frequency analysis are described as convolutions in this setting. The quantum harmonic analysis is thus a conceptual framework for the study of Gabor multipliers, and several of the results include results on Gabor multipliers as special cases.

Keywords Gabor multipliers·Tauberian theorems·Feichtinger’s algebra· Fourier–Wigner transform

Mathematics Subject Classification 47B38·47B10·35S05·42B05·43A32

1 Introduction

In time-frequency analysis, one studies a signalψ ∈ L²(R^d)by considering various time-frequency representations of ψ. An important class of time-frequency repre- sentations is obtained by fixingϕ ∈ L²(R^d)and considering theshort-time Fourier transform V_ϕψ of ψ with windowϕ, which is the function on the time-frequency planeR^2dgiven by

V_ϕψ(z)= ψ, π(z)ϕL² forz∈R^2d,

Communicated by Hans G. Feichtinger.

B

1 Department of Mathematics, NTNU Norwegian University of Science and Technology, 7491 Trondheim, Norway

where π(z) : L²(R^d) → L²(R^d)is thetime-frequency shift given byπ(z)ϕ(t) = e^2πiω·tϕ(t−x)forz=(x, ω).The intuition is thatV_ϕψ(z)carries information about the components of the signalψwith frequencyωat timex.

A question going back to von Neumann [58] and Gabor [23] is the validity of reconstruction formulas of the form

ψ=

λ∈

V_ϕψ(λ)π(λ)ξfor anyψ∈ L²(R^d), (1)

where = AZ^2d for A ∈ G L(2d,R)is a lattice in R^2d andϕ, ξ ∈ L²(R^d). It is known that (1) is indeed true for certain windowsϕ, ξ and lattices, and such formulas naturally lead to the concept ofGabor multipliers. Ifϕ, ξ ∈ L²(R^d)and m = {m(λ)}_λ∈ is a sequence of complex numbers, we define the Gabor multiplier Gm^ϕ,ξ :L²(R^d)→L²(R^d)by

Gm^ϕ,ξ(ψ)=

λ∈

m(λ)V_ϕψ(λ)π(λ)ξ.

Compared to (1) we see thatGm^ϕ,ξmodifies the time-frequency content ofψin a simple way, namely by multiplying the samples of its time-frequency representation with a maskm. Gabor multipliers have been studied in the mathematics literature by [5,9, 14,16,20,21,29,33] among others, and also in more application-oriented contributions [1,50,56].

Gabor multipliers are the discrete analogues of the much-studied localization oper- ators [2,9,10,32]. In [46] we showed that the quantum harmonic analysis developed by Werner and coauthors [39,59] provides a conceptual framework for localization operators, leading to new results and interesting reinterpretations of older results on localization operators. The goal of this paper is therefore to develop a version of quantum harmonic analysis for lattices to provide a similar conceptual framework for Gabor multipliers. Hence we continue the line of research into applications of quantum harmonic analysis from [45–47].

With this aim we introduce two convolutions of operators and sequences in Sect.4.

Following [18,40,59] we first define the translation of an operatorS on L²(R^d)by λ∈to be the operator

α_λ(S)=π(λ)Sπ(λ)^∗.

Ifc∈¹()andSis a trace class operator onL²(R^d), the convolutioncSis defined to be theoperator

cS =

λ∈

c(λ)αλ(S).

Gabor multipliers are then given by convolutions Gm^ϕ,ξ=m(ξ⊗ϕ),

whereξ⊗ϕis the rank-one operatorξ⊗ϕ(ψ)= ψ, ϕ_L²ξ. Furthermore, we define the convolutionST of two trace class operatorsSandTto be thesequenceover given by

ST(λ)=tr(Sα_λ(Tˇ)),

whereTˇ = P T P withP the parity operator Pψ(t)=ψ(−t)forψ ∈ L²(R^d). In Sect.4we investigate the commutativity and associativity of these convolutions, extend their domains and in Proposition4.3we establish a version of Young’s inequality for convolutions of operators and sequences.

An important tool throughout the paper is a Banach spaceBof trace class operators, consisting of operators with Weyl symbol in the so-called Feichtinger algebra [15].

The use ofBallows us to obtain continuity results for the convolutions with respect to ^p()and Schatten-pclasses—an important example is Proposition4.1which states that

ST¹₍₎S_BT_T

for S ∈ Band trace class T, where · _T is the trace class norm. While there are other classes of operators that would ensure that ST ∈ ¹(), see for instance the Schwartz operators [38],Bhas the advantage of being a Banach space, hence allowing the use of tools such as Banach space adjoints. The spaceBhas previously been studied by [14,17,18] among others.

To complement the convolutions, we introduce Fourier transforms of sequences and operators in Sect.5. For a sequence c ∈ ¹()we use its symplectic Fourier series

F_σ(c)(z)=

λ∈

c(λ)e^2πiσ(λ,z) forz∈R^2d,

whereσ (z,z)=ω·x −x·ω forz=(x, ω),z =(x, ω).As a Fourier transform for trace class operatorsSwe use the Fourier–Wigner transform

FW(S)(z)=e^{−πi x·ω}tr(π(−z)S) forz=(x, ω)∈R^2d.

Equipped with both convolutions and Fourier transforms, we naturally ask whether the Fourier transforms turn convolutions into products. We show in Theorem5.3for z∈R^2dthat

F_σ(ST)(z)= 1

||

λ^◦∈^◦

FW(S)(z+λ^◦)FW(T)(z+λ^◦), (2)

where^◦is the adjoint lattice ofdefined in Sect.5, and in Propositions5.4and5.5 we show that

FW(cS)(z)=F_σ(c)(z)FW(S)(z). (3)

These results include as special cases the so-called fundamental identity of Gabor analysis [19,36,52,57] and results on the spreading function of Gabor multipliers due to [14]. Equations (2) and (3) hold for general classes of operators and sequences, and we take care to give a precise interpretation of the objects and equalities in all cases.

A fruitful approach to Gabor multipliers due to Feichtinger [16] is to consider the so-called Kohn–Nirenberg symbol of operators. The Kohn–Nirenberg symbol of an operator S onL²(R^d)is a function onR^2d, and Feichtinger used this to reduce questions about Gabor multipliers in the Hilbert–Schmidt operators to questions about functions in L²(R^2d). This approach has later been used in other papers on Gabor multipliers [5,14,20]. As Gabor multipliers are examples of convolutions, we show in Sect.6that this approach can be generalized and phrased in terms of our quantum harmonic analysis, and that one of the main results of [16] finds a natural interpretation as a Wiener’s lemma in our setting—see Theorem6.3, Corollary6.3.1and the remarks following the corollary.

In Sect.7we show the extension of some deeper results of harmonic analysis on R^dto our setting. We obtain an analogue of Wiener’s classical Tauberian theorem in Theorem7.3, similar to the results of Werner and coauthors [39,59] in the continuous setting. As an example we have the following equivalent statements forS∈B: (i) The set of zeros ofF_σ(SSˇ^∗)contains no open subsets inR^2d/^◦. (ii) IfcS=0 forc∈¹(), thenc=0.

(iii) BSis weak*-dense in^∞().

These results are related to earlier investigations of Gabor multipliers by Feichtinger [16]. In particular, he showed that ifS=ξ⊗ϕis a rank-one operator andF_σ(SSˇ^∗) hasnozeros, then anym∈^∞()can be recovered from the Gabor multiplierGm^ϕ,ξ. Since Gabor multipliers are given by convolutions, the equivalence (i) ⇐⇒ (ii) shows that we can recoverm∈¹()fromGm^ϕ,ξunder the weaker condition (i)—this holds in particular for finite sequencesm.

Finally, we apply our techniques to prove a version of Wiener’s division lemma in Theorem7.4. At the level of Weyl symbols this turns out to reproduce a result by Gröchenig and Pauwels [31], but in our context it has the following interpretation:

IfFW(S)has compact support for some operatorS, and the support is sufficiently small compared to the density of, then there exists a sequencem ∈ ^∞() such thatS =mAfor some A∈ B. IfS belongs to the Schatten-pclass of compact operators, thenm∈^p().

The above result fits well into the common intuition that operatorsSwith compactly supportedFW(S)(so-called underspread operators) can be approximated by Gabor multipliers [14]—i.e. by operatorscT whereT is a rank-one operator. The result shows that if we allowT to beanyoperator inB, then any underspread operatorSis precisely of the formS=cT for a sufficiently dense lattice.

We end this introduction by emphasizing the hybrid nature of our setting. In [59], Werner introduced quantum harmonic analysis of functions onR^2dand operators on the Hilbert space L²(R^d). We are considering the discrete setting of sequences on a lattice instead of functions onR^2d. If we had modified the Hilbert spaceL²(R^d) accordingly, many of our results would follow by the arguments of [59], as already

outlined in [39]. However, we keep the same Hilbert spaceL²(R^d)as in the continuous setting. We are therefore mixing the discrete (lattices) and the continuous (L²(R^d)), which leads to some extra intricacies.

2 Conventions

By a latticewe mean a full-rank lattice inR^2d, i.e.=AZ^2dforA∈G L(2d,R).

The volume of = AZ^2d is|| := det(A). For a lattice, the Haar measure on R^2d/will always be normalized so thatR^2d/has total measure 1.

IfXis a Banach space andX its dual space, the action ofy∈ X onx∈ Xis denoted by the brackety,xX,X, where the bracket is antilinear in the second coordinate to be compatible with the notation for inner products in Hilbert spaces. This means that we are identifying the dual spaceX withantilinear functionals onX. For two Banach spacesX,Ywe useL(X,Y)to denote the Banach space of continuous linear operators fromX toY, and ifX =Y we simply writeL(X). The notationP Qmeans that there is someC>0 such thatP≤C·Q.

3 Spaces of Operators and Functions

3.1 Time-Frequency Shifts and the Short-Time Fourier Transform

Forz=(x, ω)∈R^2dwe define thetime-frequency shiftoperatorπ(z)by (π(z)ψ)(t)=e^2πiω·tψ(t−x) forψ∈L²(R^d).

Hence π(z) can be written as the composition M_ωTx of a translation operator (Txψ)(t) = ψ(t −x)and a modulation operator(M_ωψ)(t) = e^2πiω·tψ(t). The time-frequency shiftsπ(z)are unitary operators onL²(R^d). Forψ, ϕ ∈ L²(R^d)we can use the time-frequency shifts to define theshort-time Fourier transform V_ϕψof ψwith windowϕby

V_ϕψ(z)= ψ, π(z)ϕL² forz∈R^2d.

The short-time Fourier transform satisfies an orthogonality condition, sometimes called Moyal’s identity [22,27].

Lemma 3.1 [Moyal’s identity] Ifψ1, ψ2, ϕ1, ϕ2 ∈ L²(R^d), then V_ϕiψj ∈ L²(R^2d) for i,j∈ {1,2}, and the relation

V_ϕ1ψ1,V_ϕ2ψ2

L² = ψ1, ψ2L²ϕ1, ϕ2L²

holds, where the leftmost inner product is in L²(R^2d)and those on the right are in L²(R^d).

By replacing the inner product in the definition ofV_ϕψby a duality bracket, one can define the short-time Fourier transform for other classes ofψ, ϕ. The most general case we need is that of a Schwartz functionϕ ∈ S(R^d)and a tempered distribution ψ∈S(R^d); we define

V_ψϕ(z)= ψ, π(z)ϕ_S,S forz∈R^2d. 3.2 Feichtinger’s Algebra

An appropriate space of functions for our purposes will be Feichtinger’s algebra S0(R^d), first introduced by Feichtinger in [15]. To define S0(R^d), letϕ0denote the L²-normalized Gaussianϕ0(x)=2^d/4e^−πx·xforx ∈ R^d. ThenS0(R^d)is the space of allψ∈S(R^d)such that

ψS0 :=

R^2d|V_ϕ0ψ(z)|d z<∞.

With the norm above,S0(R^d)is a Banach space of continuous functions and an algebra under multiplication and convolution [15]. By [27, Thm. 11.3.6], the dual space of S0(R^d)is the spaceS₀(R^d)consisting of allψ∈S(R^d)such that

ψS₀ := sup

z∈R^2d|V_ϕ0ψ(z)|d z<∞, where an elementψ∈ S₀(R^d)acts onφ∈ S0(R^d)by

φ, ψS₀,S0 =

R^2d V_ϕ0φ(z)V_ϕ0ψ(z)d z.

We get the following chain of continuous inclusions:

S(R^d) → S0(R^d) →L²(R^d) →S₀(R^d) →S(R^d).

One important reason for using Feichtinger’s algebra is that it consists of continuous functions, and that sampling them over a lattice produces a summable sequence [15, Thm. 7C)].

Lemma 3.2 (Sampling Feichtinger’s algebra) Let be a lattice inR^2d and f ∈ S0(R^2d). Then f|= {f(λ)}_λ∈∈¹()with

f|¹ fS0, where the implicit constant depends only on the lattice.

3.3 The Symplectic Fourier Transform

We will use thesymplectic Fourier transformFσf of functions f ∈ L¹(R^2d), defined by

F_σf(z)=

R^2d f(z)e^{−2πiσ(z,z)}d z,

whereσis the standard symplectic formσ (z,z)=ω·x −x·ω forz=(x, ω),z = (x, ω).Fσis a Banach space isomorphismS0(R^2d)→S0(R^2d), extends to a unitary operatorL²(R^2d)→L²(R^2d)and a Banach space isomorphismS₀(R^2d)→S₀(R^2d) [18, Lem. 7.6.2]. In fact,F_σ is its own inverse, so that F_σ(F_σ(f)) = f for f ∈ S₀(R^2d)[11, Prop. 144].

3.4 Banach Spaces of Operators onL²(R^d)

The results of this paper concern operators on various function spaces, and we will pick operators from two kinds of spaces: the Schatten-pclassesT^pfor 1≤ p ≤ ∞ and a spaceBof operators defined using the Feichtinger algebra.

3.4.1 The Schatten Classes

Starting with the Schatten classes, we recall that any compact operatorSonL²(R^d) has a singular value decomposition [7, Remark 3.1], i.e. there exist two orthonormal sets{ψn}n∈N and{φn}n∈N inL²(R^d)and a bounded sequence of positive numbers {sn(S)}n∈Nsuch thatSmay be expressed as

S =

n∈N

sn(S)ψn⊗φn,

with convergence of the sum in the operator norm. Hereψ⊗φforψ, φ ∈ L²(R^d) denotes the rank-one operatorψ⊗φ(ξ)= ξ, φL²ψ.

For 1≤ p<∞we define theSchatten-p classT^pof operators onL²(R^d)by T^p= {T compact: {sn(T)}n∈N∈^p}.

To simplify the statement of some results, we also defineT^∞=L(L²)with · _T^∞ given by the operator norm. The Schatten-pclassT^pis a Banach space with the norm S_T^p =

n∈Nsn(S)^p 1/p

. Of particular interest is the spaceT :=T¹; the so-called trace class operators. Given an orthonormal basis{en}n∈NofL²(R^d), the trace defined by

tr(S)=

n∈N

Sen,enL²

is a well-defined and bounded linear functional onT, and independent of the orthonor- mal basis{en}n∈Nused. The dual space ofT isL(L²)[7, Thm. 3.13], andT ∈L(L²) defines a bounded antilinear functional onT by

T,S_L(L²),T =tr(T S^∗) forS ∈T.

Another special case is the space of Hilbert–Schmidt operatorsHS :=T², which is a Hilbert space with inner product

S,T_HS=tr(ST^∗).

3.4.2 The Weyl Transform and Operators with Symbol inS0(R^2d)

The other class of operators we will use will be defined in terms of theWeyl transform.

We first need thecross-Wigner distribution W(ξ, η)of two functionsξ, η∈L²(R^d), defined by

W(ξ, η)(x, ω)= R^dξ

x+ t

2

η

x−t 2

e^−2πiω·tdt for(x, ω)∈R^2d.

For f ∈ S₀(R^2d), we define theWeyl transform Lf of f to be the operatorLf : S0(R^d)→S₀(R^d)given by

Lfη, ξ

S₀,S0 := f,W(ξ, η)S₀,S0 for anyξ, η∈S0(R^d).

f is called the Weyl symbolof the operator Lf. By the kernel theorem for modu- lation spaces [27, Thm. 14.4.1], the Weyl transform is a bijection from S₀(R^2d)to L(S0(R^d),S₀(R^d)).

Notation In particular, anyS ∈ L(S0(R^d),S₀(R^d))has a Weyl symbol, and we will denote the Weyl symbol ofSbyaS. By definition, this means thatLaS =S.

It is also well-known that the Weyl transform is a unitary mapping fromL²(R^2d)to HS[49]. This means in particular that

S,T_HS= aS,aTL² forS,T ∈HS,

which often allows us to reduce statements about Hilbert–Schmidt operators to state- ments aboutL²(R^2d).

We then defineBto be the Banach space of continuous operatorsS : S0(R^d)→ S₀(R^d)such thataS∈S0(R^2d), with norm

S_B:= aSS0.

Bconsists of trace class operatorsL²(R^d)and we have a norm-continuous inclusion ι:B→T [25,30].

Example 3.1 Ifφ, ψ ∈L²(R^d), consider the rank-one operatorφ⊗ψ.Its Weyl symbol is the cross-Wigner distributionW(φ, ψ)[11, Cor. 207], andW(φ, ψ)∈ S0(R^2d)if and only ifφ, ψ ∈ S0(R^d)[11, Prop. 365]. The simplest examples of operators inB are thereforeφ⊗ψforφ, ψ ∈S0(R^d).

The dual space B can also be identified with a Banach space of operators. By definition,τ :B→S0(R^2d)given byτ(S)=aSis an isometric isomorphism. Hence the Banach space adjointτ^∗:S₀(R^2d)→B is also an isomorphism. Since the Weyl transform is a bijection fromS₀(R^2d)toL(S0(R^d),S₀(R^d)), we can identifyB with operatorsS0(R^d)→ S₀(R^d):

B ←−→^τ∗ S₀(R^2d) Weyl calculus

←−−−−−→L(S0(R^d),S₀(R^d)).

In this paper we will always consider elements ofB as operatorsS0(R^d)→S₀(R^d) using these identifications. SinceL(L²)is the dual space ofT, the Banach space adjoint ι^∗:L(L²)→B is a weak*-to-weak*-continuous inclusion ofL(L²)intoB. Remark For more results onBandB we refer to [17,18]. In particular we mention that we could have definedBusing other pseudodifferential calculi, such as the Kohn–

Nirenberg calculus, and still get the same spaceBwith an equivalent norm. We would also like to point out that the statements of this section may naturally be rephrased using the notion of Gelfand triples, see [18].

3.5 Translation of Operators

The idea of translating an operatorS ∈ L(L²)byz ∈ R^2d using conjugation with π(z)has been utilized both in physics [59] and time-frequency analysis [18,40]. More precisely, we define forz∈R^2dandS∈B the translation ofSbyzto be the operator

αz(S)=π(z)Sπ(z)^∗.

We will also need the operation S → ˇS = P S P, where P is the parity operator (Pψ)(t)=ψ(−t)forψ∈L²(R^d). The main properties of these operations are listed below, note in particular that part(i)supports the intuition thatαz is a translation of operators. See Lemmas 3.1 and 3.2 in [46] for the proofs.

Lemma 3.3 Let S∈B.

(i) If aSis the Weyl symbol of S, then the Weyl symbol ofαz(S)is Tz(aS).

(ii) αz(αz(S))=αz+z(S).

(iii) The operationsαz,^∗andˇare isometries onB,B andT^pfor1≤ p≤ ∞.

(iv) (S^∗)q=(S)ˇ ^∗.

By the last part we can unambiguously writeSˇ^∗.

4 Convolutions of Sequences and Operators

In [59], the convolution of a function f ∈L¹(R^2d)and an operatorS∈T was defined by the operator-valued integral

fS=

R^2d f(z)αz(S)d z

and the convolution of two operatorsS,T ∈T was defined to be thefunction ST(z)=tr(Sαz(Tˇ)) forz∈R^2d.

These definitions, along with a Fourier transform defined for operators, have been shown to produce a theory of quantum harmonic analysis with non-trivial conse- quences for topics such as quantum measurement theory [39] and time-frequency analysis [46]. The setting whereR^2d is replaced by some lattice ⊂ R^2d is fre- quently studied in time-frequency analysis, and our goal is therefore to develop a theory of convolutions and Fourier transforms of operators in that setting.

For a sequencec∈¹()andS∈T, we define the operator cS:=Sc:=

λ∈

c(λ)α_λ(S), (4)

and for operatorsS∈BandT ∈T we define the sequence

ST(λ)=ST(λ) forλ∈. (5)

HenceST is thesequenceobtained by restricting thefunction ST to.

Remark We use the same notationfor the convolution of an operator and a sequence and for the convolution of two operators. The correct interpretation ofwill always be clear from the context.

Since α_λ is an isometry on T and B, cS is well-defined with cS_T ≤ c¹S_T forS∈T and similarlycS_B ≤ c¹S_B forS∈B. The fact that ST is a well-defined and summable sequence onis less straightforward.

Proposition 4.1 If S ∈ B and T ∈ T, then ST ∈ ¹() with ST¹ S_BT_T.

Proof By [46, Thm. 8.1] we know thatST ∈ S0(R^2d)withSTS₀ S_BT_T. Hence the result follows from Lemma3.2andST(λ)=ST(λ).

4.1 Gabor Multipliers and Sampled Spectrograms

If we consider rank-one operators, these convolutions reproduce well-known objects from time-frequency analysis. First consider the rank-one operatorξ1⊗ξ2forξ1, ξ2∈

L²(R^d). The operatorsc(ξ1⊗ξ2)are well-known in time-frequency analysis as Gabor multipliers[5,14,16,20]: it is simple to show that

αλ(ξ1⊗ξ2)=(π(λ)ξ1)⊗(π(λ)ξ2),

so ifc∈¹()it follows from the definition (4) thatc(ξ1⊗ξ2)acts onψ∈L²(R^d) by

c(ξ1⊗ξ2)ψ =

λ∈

c(λ)Vξ2ψ(λ)π(λ)ξ1, (6)

which is the definition of the Gabor multiplierG^ξc^2,ξ1 used in time-frequency analysis [20], i.e.Gc^ξ2,ξ1 =c(ξ1⊗ξ2).

Remark In this sense, operators of the formcSare a generalization of Gabor mul- tipliers. We mention that this is a different generalization from themultiple Gabor multipliersintroduced in [14].

If we pick another rank-one operatorϕˇ1⊗ ˇϕ2forϕ1, ϕ2∈ L²(R^d)(hereϕ(t)ˇ =ϕ(−t)), one can calculate using the definition (5) that

(ξ1⊗ξ2)(ϕˇ1⊗ ˇϕ2)(λ)=V_ϕ2ξ1(λ)V_ϕ1ξ2(λ). (7) In particular, ifϕ1=ϕ2=ϕandξ1=ξ2=ξ, then

(ξ⊗ξ)(ϕˇ⊗ ˇϕ)(λ)= |V_ϕξ(λ)|². (8) The function|V_ϕξ(z)|²is the so-called spectrogram ofξwith windowϕ, hence(ξ⊗ ξ)(ϕˇ⊗ ˇϕ)consists of samples of the spectrogram over.

Finally, ifS∈T is any operator, then one may calculate that

Sϕˇ1⊗ ˇϕ2(λ)= Sπ(λ)ϕ1, π(λ)ϕ2_L², (9) often called the lower symbol ofSwith respect toϕ1, ϕ2and[16].

Remark In particular, Proposition4.1does not hold for allS∈T. By Remark 4.6 in [5], there exists a functionψ∈L²(R)such that

(m,n)∈Z²

(ψ⊗ψ)_Z²(ψˇ ⊗ ˇψ)(m,n)=

(m,n)∈Z²

|V_ψψ(m,n)|²= ∞.

Sinceψ⊗ψ,ψˇ ⊗ ˇψ ∈T, this shows that the assumptionS ∈Bin Proposition4.1 is necessary.

4.2 Associativity and Commutativity of Convolutions

Since the convolutionSTof two operatorsS,T ∈T is commutative in the continuous setting [59, Prop. 3.2], it follows from the definitions that the convolutions (4) and (5) are commutative. It is also a straightforward consequence of the definitions that the convolutions are bilinear.

In the original theory of Werner [59], the associativity of the convolution operations is of fundamental importance. Associativity still holds in some cases when moving fromR^2d to, but we will later see in Corollary7.2.2that the convolution of three operators over a lattice is not associative in general. In what follows,c∗ddenotes the usual convolution of sequences

c∗d(λ)=

λ∈

c(λ)d(λ−λ).

Proposition 4.2 (Associativity)Let c,d ∈¹(), S∈Band T ∈T. Then (i) c∗(ST)=(cS)T ,

(ii) (c∗d)T =c(dT).

Proof For the proof of(i), we write out the definitions of the convolutions and use the commutativityST =TSto get

c∗(ST)(λ)=c∗(TS)(λ)

=

λ∈

c(λ)tr(Tα_λ−λ(Sˇ))

=tr T

λ∈

c(λ)α_λ−λ(Sˇ)

=tr Tαλ

λ∈

c(λ)α_−λ(Sˇ)

by Lemma3.3

=tr Tαλ P

λ∈

c(λ)α_λ(S)P

=T(cS) by (4) and (5)

=(cS)T by commutativity.

We have used the easily checked relationα_−λ(S)ˇ = Pα_λ(S)P. For the second part, we find that

(c∗d)T =

λ∈

(c∗d)(λ)α_λ(T)

=

λ∈

λ∈

c(λ)d(λ−λ)α_λ(T)

=

λ∈

c(λ)

λ∈

d(λ−λ)α_λ(T)

=

λ∈

c(λ)α_λ(dT)=c(dT).

To pass to the last line we have used the relationα_λ(dT)=

λd(λ−λ)α_λ(T),

which is easily verified.

Remark Part(ii)of this result along with the trivial estimatecT_T ≤ c¹T_T shows thatT is aBanach module(see [24]) over ¹()if we define the action of c∈¹()onT ∈T bycT. The same proofs also show that this is true whenT is replaced byBor any Schatten classT ^pfor 1≤ p≤ ∞.

Example 4.1 Letϕ, ξ ∈L²(R^d)andc∈¹(), and defineS=ξ⊗ξandT = ˇϕ⊗ ˇϕ.

If we use (8) to simplify ST and (9) to simplify(cS)T, the first part of the result above becomes

c∗|V_ϕξ|²(λ)= (cξ⊗ξ)π(λ)ϕ, π(λ)ϕL². (10) In words, the convolution of a sequencecwith samples of a spectrogram|V_ϕξ|²can be described using the action of a Gabor multiplierc(ξ⊗ξ). In applications of con- volutional neural networks to audio processing, one often considers the spectrogram of an audio signal as the input to the network. Convolutions of sequences with samples of spectrograms therefore appear naturally in such networks, and the connection (10) has been exploited in this context—see the proof of [13, Thm. 1].

4.3 Young’s Inequality

The convolutions in (4) and (5) can be defined for more general sequences and operators by establishing a version of Young’s inequality [27, Thm. 1.2.1]. In the continuous case such an inequality was established by Werner [59] using theL^p-norms of functions and Schatten-p-norms of operators. In the discrete case, it is not always possible to use the Schatten-p-norms, since Proposition4.1requiresS ∈ B. We will therefore always require that one of the operators belongs toB.

A Young’s inequality for Schatten classes can then be established by first extending the domains of the convolutions by duality. If S ∈ Bandc ∈ ^∞(), we define cS∈L(L²)by

cS,R_L(L²),T :=

c,RSˇ^∗

^∞,¹ for any R∈T. (11) and ifS∈BandT ∈L(L²)=T^∞we defineTS∈^∞()by

TS,c∞(),¹():=

T,cSˇ^∗

L(L²),T for anyc∈¹(). (12)

It is a simple exercise to show that these definitions define elements ofL(L²)and ^∞() satisfyingcS_L(L²) c^∞S_B andTS^∞ ≤ T_L(L²)S_B, and that they agree with (4) and (5) whenc∈¹()orT ∈T. A standard (complex) interpolation argument then gives the following result, since (¹(), ^∞())θ = ^p()and(T¹,T^∞)θ =T^pwith ¹_p =1−θ[6]. For Gabor multipliers the second part of this result is well-known [20, Thm. 5.4.1], and a weaker version of the first part is known for p=1,2,∞[20, Thm. 5.8.3].

Proposition 4.3 (Young’s inequality) Let S∈Band1≤ p≤ ∞.

(i) If T∈T^p, thenTS^p T_T^pS_B. (ii) If c∈^p(), thencS_T^p c^pS_B.

Remark If 1∈ ^∞()is given by 1(λ)=1 for anyλ, then Feichtinger observed in [16, Thm. 5.15] thatφ∈S0(R^d)generates a so-called tight Gabor frame if and only if the Gabor multiplier 1(φ⊗φ)is the identity operatorIinL(L²). A similar result holds in the more general case: ifS∈B, then 1S^∗S=I if and only ifSgenerates a tightGabor g-frame, recently introduced in [54].

We may also use duality to define the convolution TS ∈ ^∞()of S ∈ B with T ∈B by

TS,c∞,¹ :=

T,cSˇ^∗

B,B for anyc∈¹(), (13) which agrees with (12) when T ∈ L(L²) ⊂ B and satisfies ST^∞ ≤ S_BT_B. We end this section by showing that the spacec0()of sequences van- ishing at infinity corresponds to compact operators under convolutions with S ∈ B.

The second part of this statement is due to Feichtinger [16, Thm. 5.15] for the special case of Gabor multipliers.

Proposition 4.4 Let S ∈ B. If T is a compact operator, then TS ∈ c0(). If c∈c0(),then cS is a compact operator on L²(R^d).

Proof By [46, Prop. 4.6], thefunction TSbelongs to the spaceC0(R^2d)of continuous functions vanishing at infinity. SinceTSis simply the restriction ofTS to, it follows thatTS∈c0(). For the second part, letcNbe the sequence

cN(λ)=

c(λ) if |λ|<N

0 otherwise.

ThencNS =

|λ|<Nc(λ)αλ(S)is a compact operator for each N ∈ N, and by Proposition4.3and the bilinearity of convolutions

cS−cNS_L(L²)≤ c−cN^∞S_B→0 asN → ∞.

HencecSis the limit in the operator topology of compact operators, and is therefore

itself compact.

5 Fourier Transforms

In [59], Werner observed that if one defines a Fourier transform of an operatorS ∈T to be the function

FW(S)(z):=e^{−πi x·ω}tr(π(−z)S) forz=(x, ω)∈R^2d, then the formulas

FW(fS)=Fσ(f)FW(S), Fσ(ST)=FW(S)FW(T) (14) hold for f ∈ L¹(R^2d)andS,T ∈T. The transformFW, called theFourier–Wigner transform (or the Fourier-Weyl transform [59]) is an isomorphism FW : B → S0(R^2d), can be extended to a unitary mapFW : HS → L²(R^2d), and to an iso- morphismFW : B → S₀(R^2d)by definingFW(S)forS ∈ B by duality [18, Cor.

7.6.3]:

FW(S),fS₀,S0 := S, ρ(f)_B,B for any f ∈ S0(R^2d). (15) Hereρ:S0(R^2d)→Bis the inverse ofFW. In fact,FW and the Weyl transform are related by a symplectic Fourier transform: for anyS∈B we have

FW(S)=F_σ(aS),

whereaSis the Weyl symbol ofS. As an important special case, the Fourier–Wigner transform of a rank-one operatorφ⊗ψis

FW(φ⊗ψ)(x, ω)=e^{πi x·ω}V_ψφ(x, ω). (16) Since we have defined convolutions of operators and sequences, it is natural to ask whether a version of (14) holds in our setting. We start by defining a suitable Fourier transform of sequences.

Symplectic Fourier Series

For the purposes of this paper, we identify the dual groupR^2dwithR^2dby the bijection R^2d z → χz ∈R^2d, whereχz is thesymplecticcharacter¹χz(z) = e^2πiσ(z,z). Given a lattice⊂R^2d, it follows that the dual group ofis identified withR^2d/^◦

1 Phase space, which in this paper isR^2d, is more properly described by (the isomorphic) spaceR^d×R^d. The symplectic characters appear because they are the natural way of identifying the groupR^d×R^dwith its dual group.

(see [12, Prop. 3.6.1]), where^◦is the annihilator group ^◦= {λ^◦∈R^2d:χλ^◦(λ)=1 for anyλ∈}

= {λ^◦∈R^2d:e^{2πiσ(λ◦,λ)}=1 for anyλ∈}.

The group^◦is itself a lattice, namely the so-calledadjoint latticeoffrom [18,52].

Given this identification of the dual group of, the Fourier transform ofc∈ ¹() is the symplectic Fourier series

F_σ(c)(˙z):=

λ∈

c(λ)e^2πiσ(λ,z).

Here˙zdenotes the image ofz∈R^2dunder the natural quotient mapR^2d →R^2d/^◦, soF_σ(c)is a function onR^2d/^◦. If we denote byA(R^2d/^◦)the Banach space of functions onR^2d/^◦with symplectic Fourier coefficients in¹(), the Feichtinger algebra has the following property [15, Thm. 7 B)].

Lemma 5.1 Ifis a lattice, the periodization operator P:S0(R^2d)→ A(R^2d/) defined by

P(f)(˙z)= ||

λ∈

f(z+λ) for z∈R^2d

is continuous and surjective.

Remark (i) Since|^◦| = _||¹ [18, Lem. 7.7.4], we have P◦(f)(z)˙ = 1

||

λ^◦∈

f(z+λ^◦).

(ii) One may define Feichtinger’s algebra S0(G)for any locally compact abelian groupG [15]. In fact, all our function spaces besides L²(R^d)are examples of Feichtinger’s algebra, sinceS0()=¹()andS0(R^2d/^◦)=A(R^2d/^◦).

When we identify the dual group ofwithR^2d/^◦, the Poisson summation formula for functions inS0(R^2d)takes the following form.

Theorem 5.2 (Poisson summation) Letbe a lattice inR^2d and assume that f ∈ S0(R^2d). Then

1

||

λ^◦∈^◦

f(z+λ^◦)=

λ∈

F_σ(f)(λ)e^2πiσ(λ,z)for z∈R^2d.

Proof This is [12, Thm. 3.6.3] withA=R^2d,B=^◦and using(^◦)^◦=.To get equality for anyz∈R^2d, we use that

λ^◦∈^◦ f(z+λ^◦)defines a continuous function

onR^2d/^◦by Lemma5.1.

SinceF_σis a Fourier transform it extends to a unitary mappingF_σ : ²() → L²(R^2d/^◦)satisfying

F_σ(c∗d)=F_σ(c)F_σ(d) (17) forc∈¹()andd ∈²().

5.1 The Fourier Transform ofS₃T

We now consider a version of (14) for sequences. The formula forF_σ(ST)is a simple consequence of the Poisson summation formula.

Theorem 5.3 Let S∈Band T ∈T. Then F_σ(ST)(˙z)= 1

||

λ^◦∈^◦

FW(S)(z+λ^◦)FW(T)(z+λ^◦)

=P◦(FW(S)FW(T))(˙z) for any z∈R^2d.

Proof From [46, Thm. 8.2], we know that ST ∈ S0(R^2d). Hence Fσ(ST) = FW(S)FW(T) ∈ S0(R^2d)sinceFσ : S0(R^2d) → S0(R^2d)is an isomorphism. By applying Poisson’s summation formula from Theorem5.2to f =FW(S)FW(T), we find that

1

||

λ^◦∈^◦

FW(S)(z+λ^◦)FW(T)(z+λ^◦)=

λ∈

Fσ(FW(S)FW(T))(λ)e^2πiσ(λ,z)

=

λ∈

ST(λ)e^2πiσ(λ,z),

where we used thatFσ is its own inverse to conclude that

Fσ(FW(S)FW(T))(λ)=Fσ(Fσ(ST))(λ)=ST(λ)=ST(λ).

SinceFW(S)FW(T) ∈ S0(R^2d), Theorem5.2says that the equation holds for any

z∈R^2d.

Remark Theorem5.3has also been proved and used in [44, Cor. A.3] in noncommu- tative geometry, with stronger assumptions onS,T.

Theorem5.3has many interesting special cases. We will frequently refer to the fol- lowing version, which follows since a short calculation using the definition of the Fourier–Wigner transform shows that

FW(Sˇ^∗)(z)=FW(S)(z). (18)