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REPRESENTATION OF LINEAR OPERATORS BY GABORMULTIPLIERS

PETER C. GIBSON, MICHAEL P. LAMOUREUX, AND GARY F. MARGRAVE

Abstract. We consider a continuous version of Gabor multipliers: operators con-sisting of a short-time Fourier transform, followed by multiplication by a distribu-tion on phase space (called the Gabor symbol), followed by an inverse short-timeFourier transform, allowing different localizing windows for the forward and inversetransforms. For a given pair of forward and inverse windows, which linear operatorscan be represented as a Gabor multiplier, and what is the relationship between the(non-classical) Kohn-Nirenberg symbol of such an operator and the correspond-ing Gabor symbol? These questions are answered completely for a special class of“compatible” window pairs. In addition, concrete examples are given of windowsthat, with respect to the representation of linear operators, are more general thanstandard Gaussian windows. The results in the paper help to justify techniques de-veloped for seismic imaging that use Gabor multipliers to represent nonstationaryfilters and wavefield extrapolators.

[Submitted to the Journal of Fourier Analysis and Applications, 19 Sept. 2003]

1. Introduction

This paper is concerned with the representation of linear operators by Gabor mul-tipliers. For us a Gabor multiplier is an operator of the form: a short-time Fouriertransform, followed by multiplication by a distribution on phase space—called theGabor symbol—followed by an inverse short-time Fourier transform. We explicitlyallow the analysis window, upon which the forward short-time Fourier transform isbased, to be different from the synthesis window, which serves as the basis for theinverse short-time Fourier transform.

More precisely, we study the representation of general (i.e., non-classical) Kohn-Nirenberg pseudodifferential operators—see for example [4, Section 14.1]—by meansof Gabor multipliers. As is well-known, the class of general Kohn-Nirenberg operatorsis simply all linear operators (essentially this is a version of the Schwartz kerneltheorem—see (5) in Section 2.1). However, our aim is to analyze the correspondencebetween the Kohn-Nirenberg symbol, which in the general setting is an arbitrarydistribution, and the Gabor symbol, of a given operator, provided the latter exists.The Gabor symbol of an operator depends on a choice of analysis and synthesiswindows; indeed for a fixed window pair, there may be a limited class of linearoperators that may be represented as a Gabor multiplier.

We are grateful for the stimulating atmosphere of the Banff International Research Station, wherepart of this research was carried out during the Workshop on Scattering and Inverse Scattering,March 2003.

1

2 Peter C. Gibson, Michael P. Lamoureux, and Gary F. Margrave

As far as we know the problem of representing Kohn-Nirenberg operators exactlyby (our version of) Gabor multipliers is new, and in this paper we confine ourselvesto the following two fundamental questions.

(1) For a given pair of analysis and synthesis windows, which Kohn-Nirenbergoperators may be represented as a Gabor multiplier?

(2) Are there explicit formulas relating the Gabor and Kohn-Nirenberg symbols?

Much of the paper rests on a key technical notion of “compatible window pair”. Forsuch windows we have a complete answer to both of the above questions. A commonwindow choice for the short-time Fourier transform is a Gaussian, both as analysisand synthesis windows, and this serves as the prototype of a compatible windowpair. One concrete result that emerges from our analysis of compatible windowsis a new type of window, which we call an “extreme value” window—an analytic,rapidly decreasing window, with analytic, rapidly decreasing Fourier transform. Itis more general than a Gaussian in the sense that the class of operators that can berepresented as a Gabor multiplier based on Gaussian windows is a proper subset ofthe class that can be represented using extreme value windows.

The basic questions that we address here are motivated by a particular applica-tion, seismic imaging, where pseudodifferential operators play an important role [6, 7]and Gabor multipliers have been successfully implemented as an efficient and flexiblemethod of seismic deconvolution [8]. These implementations rest on the fact thatthere exist efficient numerical schemes, described in [5], by which to evaluate dis-cretized versions of Gabor multipliers. But something that has been lacking so faris a theoretical analysis of the relationship between the original operators—i.e. ex-trapolators or filters—which are easily expressed as Kohn-Nirenberg operators, andthe corresponding continuous Gabor multipliers that have been used to stand in forthem. The results presented here serve to fill in the gap.

There is an important issue concerning the evaluation of linear operators in prac-tical applications that we do not address in the present paper, namely, to quantifythe degree of approximation to a continuous Gabor multiplier that is attained bydiscretized versions of it. For results in this direction, see, for example, [1, Theo-rem 5.9.6] and [9, Theorem 4.2.1].

The paper is organized as follows. The mathematical framework in which weformulate the problems described above is made precise in Section 2.1, where we fixthe basic notational conventions. Theorem 1 of Section 2.2 gives a straightforwardway to express the Schwartz kernel of a linear operator in terms of its Gabor symbol,which in turn yields an expression for the Kohn-Nirenberg symbol in terms of theGabor symbol. The converse problem of expressing the Gabor symbol in terms of theKohn-Nirenberg symbol seems to be much more difficult, and takes up the bulk of thepaper. Section 2.3 provides the basic motivation for our key technical definition ofcompatible windows in Section 3.1. An alternate definition of compatibility, provedin Theorem 2 of Section 3.2 to be equivalent to the original, leads to a sequence oftechnical results in Section 3.2 that lay the groundwork for our main Theorems. Theseare Theorem 3 and Theorem 4 of Section 3.3. In Section 3.4 we apply Theorem 4 to

Submitted to the Journal of Fourier Analysis and Applications, 19 September 2003

Representation of linear operators by Gabor multipliers 3

show that extreme value windows serve to represent a wider class of linear operatorsas Gabor multipliers than is possible with Gaussian windows. We discuss briefly inSection 3.5 a natural extension of the notion of compatibility to singular windows.Lastly, Section 4 provides a short summary of the paper.

2. Preliminaries

2.1. Notation and conventions. Our notation is mostly standard, with the excep-tions that: (i) we use the version of the Fourier transform that has a factor of 2πin the exponent, and (ii) we take tempered distributions to be continuous, conjugatelinear, rather than linear, functionals on the space of Schwartz class functions. Gen-erally speaking, we deal with functions and distributions on Rn, where the value of nis fixed within a given context, and Rn is always the domain of integration, which weomit. We indicate a point in R2n by a pair (x, y) of points x, y ∈ Rn, and integrationover R2n is indicated by a pair of integral signs. For convenient reference we havecompiled in Table 1 a list of some of the function spaces and operators that we userepeatedly.

We work within the basic framework of Ln, the space of continuous, linear operators

L : Sn → S ′n,

making frequent use of the correspondence between Ln and S ′2n [10]. More precisely,

given a linear operator L ∈ Ln, there is a unique distribution K = K(L) ∈ S ′2n, its

Schwartz kernel, that satisfies the equation

(1) 〈K, θ ⊗ ϕ〉 = 〈Lϕ, θ〉 ∀ϕ, θ ∈ Sn.And conversely, given K ∈ S ′

2n, the equation (1) evidently determines a uniqueL ∈ Ln. (Here ⊗ denotes the tensor product: f ⊗ g(x, y) = f(x)g(y).) The adjointof a linear operator L ∈ Ln is the linear operator L∗ ∈ Ln defined by the equation

(2) 〈L∗ϕ, θ〉 = 〈Lθ, ϕ〉 ∀ϕ, θ ∈ Sn.In order for the operator Tψ, listed in Table 1, to be well-behaved, some restrictions

have to be placed on ψ; in this regard we introduce the notion of a “tempered changeof variables”, as follows.

Definition 1. We say that a smooth, invertible map

ψ : Rm → Rm,

is a tempered change of variables if each of the operators Tψ, T ∗ψ , Tψ−1 , T ∗

ψ−1 mapsSm into Sm.

There is one tempered change of variables on R2n whose corresponding operatorwe assign a special notation:

(3) T− = Tψ− , where ψ−(x, y) = (x, y − x).

Given an arbitrary tempered distribution σ(x, ξ) ∈ S ′2n, we write σ(X,D) for the

operator defined by the formula

(4) σ(X,D) : Sn → S ′n; σ(X,D)ϕ(x) =

∫e2πix·ξσ(x, ξ)Fϕ(ξ) dξ.



Table 1. Notation

Function and distribution spaces

Sn Schwartz class functions ϕ : Rn → Rn

S ′n (conjugate linear) tempered distributions u : Sn → C

PnC∞ functions ϕ : Rn → Rn such that |∂αϕ| isbounded by a polynomial pα for every multi-index α

Operators

Description Symbol Action on functions Adjoint

Compositionwith a changeof variablesψ : Rm → Rm

Tψ ϕ 7→ ϕ ◦ ψ T ∗ψ = |det Jψ−1| Tψ−1

Fouriertransform

F : Sn → Sn(S ′

n → S ′n)

ϕ(x) 7→ ϕ(ξ)=∫e−2πiξ·xϕ(x) dx

F∗ = F−1

PartialFouriertransform

F1 : S2n → S2n

(S ′2n → S ′

2n)ϕ(x, y) 7→ F1ϕ(ξ, y)=∫e−2πiξ·xϕ(x, y) dx

F∗1 = F−1

1

PartialFouriertransform

F2 : S2n → S2n

(S ′2n → S ′

2n)ϕ(x, y) 7→ F1ϕ(x, η)=∫e−2πiη·yϕ(x, y) dy

F∗2 = F−1

2

ModulationMξ : Sn → Sn

(S ′n → S ′

n)ϕ(x) 7→ e2πiξ·xϕ(x) M∗

ξ = M−ξ

TranslationTx : Sn → Sn

(S ′n → S ′

n)ϕ(t) 7→ ϕ(t− x) T ∗

x = T−x

Multiplicationby a ∈ S ′

mNa : Pm → S ′

m ϕ 7→ aϕ —

(Here X : Rn → Rn denotes the identity map, so that, for example, Xα(x) = xα.D stands for the differential operator D = 1

2πi∂.) We refer to σ(X,D) as a Kohn-

Nirenberg pseudodifferential operator ; the distribution σ(x, ξ) is its Kohn-Nirenbergsymbol.

In the context of classical pseudodifferential operators, where the symbol σ(x, ξ)is required to be smooth and of bounded growth, the integral on the right-hand sideof (4) is inherently well-defined. A simple way to give the integral an unambiguousinterpretation in the present much more general setting is to define σ(X,D) in termsof its Schwartz kernel:

(5) K(σ(X,D)) = T−F2σ.

Since each of the operators T− and F2 carries S ′2n bijectively onto itself, it is evident

from the representation (5) that the class of Kohn-Nirenberg pseudodifferential oper-ators on Rn is identical with Ln itself. Among the various ways to represent a linear



operator, however, the Kohn-Nirenberg symbol and accompanying formal represen-tation (4) are of particular interest since, from the physical point of view, they arenatural both for partial differential operators and for nonstationary filters [4, §14.2].In other words, in applications one is sometimes given the Kohn-Nirenberg symbolof a linear operator directly.

Note that the Kohn-Nirenberg symbol σL and the Schwartz kernel K(L) of a linearoperator L : Sn → S ′

n both belong to S ′2n. But there is a sense in which these are

distinct versions of S ′2n, in that σL is a distribution on phase space, Rn × Rn, while

K(L) is a distribution on the cross product Rn × Rn of the underlying space withitself. Indeed, since it is sometimes useful to make this distinction between spaceand frequency, we reserve ξ and η for frequency variables, using other letters (suchas x, y, τ, t, v, w) for spacial variables.

2.2. The Schwartz kernel of a general Gabor multiplier. If g : Rn → C ismeasurable and bounded by a polynomial, then the formula

Vgϕ(x, ξ) = 〈MξTxg, ϕ〉defines a short-time Fourier transform Vg : Sn → P2n with analysis window g. Thebasic theory of short-time Fourier transforms is described in [4, §3]. For γ ∈ Sn, therange of Vγ lies in S2n; the adjoint V ∗

γ of Vγ is the map

V ∗γ : S ′

2n → S ′n; 〈V ∗

γ u, ϕ〉 = 〈u, Vγϕ〉.If the distribution u ∈ S ′

2n happens to be an L2 function, then V ∗γ u is given by the

formula

V ∗γ u(t) =

∫∫u(x, ξ)MξTxγ(t) dxdξ.

Definition 2. Let g : Rn → C be measurable and bounded by a polynomial, and letγ ∈ Sn. If in addition 〈g, γ〉 6= 0, then we say that each of (g, γ) and (γ, g) is awindow pair on Rn.

It is a basic fact about the short-time Fourier transform that for any window pair(g, γ), the map

(6)1

〈g, γ〉V ∗γ Vg : Sn → Sn

is the identity. Recall from Table 1 that we use the symbolNa to denote multiplicationby a.

Definition 3. Given a window pair (g, γ) on Rn and a distribution a ∈ S ′2n, we call

Mg,γa =

1

〈g, γ〉V ∗γ NaVg

a Gabor multiplier; we refer to the distribution a as its Gabor symbol.

(Note that Feichtinger and Nowak [1] use the term “short-time Fourier transformmultiplier” for a Gabor multiplier based on identical windows (g, g), while in [1]“Gabor multiplier” refers to a more general object than we have defined.) A Gabormultiplier, in the sense of Definition 3, is a linear operator belonging to Ln. Its



adjoint is also a Gabor multiplier, and the precise connection between the two worksout as follows.

Proposition 4. For any window pair (g, γ) and any distribution a ∈ S2n, the adjointof the Gabor multiplier Mg,γ

a is (Mg,γa )∗ = Mγ,g

a .

Roughly speaking, the defining structure of a Gabor multiplier means that it carriesan implicit diagonalization on phase space. From the theoretical point of view, thisfact makes it desirable to express, if possible, a given linear operator L ∈ Ln asa Gabor multiplier, the structure of the operator then being encoded in its Gaborsymbol. It is also desirable to express an operator as a Gabor multiplier from thepoint of view of applications, since there exist fast computational methods to evaluatediscretized Gabor multipliers [5].

The main problem that we are concerned with in the present paper is to expressa given Kohn-Nirenberg pseudodifferential operator as a Gabor multiplier. Beforeconsidering the issue in detail, we deal briefly with the converse problem, of expressinga given Gabor multiplier as a pseudodifferential operator. In light of the expression(5) for the Schwartz kernel of a pseudodifferential operator, the latter problem isequivalent to computing the Schwartz kernel of a Gabor multiplier. This turns outto be relatively straightforward, and can be carried out in full generality. In statingthe basic result we make use of the following notation. Let S : S4n → S2n denote themap defined by

Sρ(x, ξ) = ρ(x, x, ξ,−ξ).The corresponding adjoint is the map

S∗ : S ′2n → S ′

4n; 〈S∗u, ϕ〉 = 〈u, Sϕ〉.

Theorem 1. An arbitrary Gabor multiplier Mg,γa has Schwartz kernel

(7) K(Mg,γa ) =

1

〈g, γ〉V ∗γ⊗gS

∗a.

Proof. Note that

〈Mg,γa ϕ, θ〉 =

1

〈g, γ〉〈a, VgϕVγθ〉,

and

VgϕVγθ(x, ξ) =

∫ϕ(t)e−2πit·ξg(t− x) dt

∫θ(τ)e−2πiτ ·ξγ(τ − x) dτ

=

∫∫ϕ(t)e2πit·ξg(t− x)θ(τ)e−2πiτ ·ξγ(τ − x) dτdt

=

∫∫e2πi(t−τ)·ξγ(τ − x)g(t− x)θ⊗ϕ(τ, t) dτdt(8)

=

∫∫e−2πi(τ,t)·(ξ,−ξ)T(x,x)γ ⊗ g(τ, t)θ⊗ϕ(τ, t) dτdt

= Vγ⊗gθ⊗ϕ(x, x, ξ,−ξ).



Thus,

〈Mg,γa ϕ, θ〉 =

1

〈g, γ〉〈a, SVγ⊗gθ⊗ϕ〉

=1

〈g, γ〉〈V ∗

γ⊗gS∗a, θ⊗ϕ〉.

Since generalized Kohn-Nirenberg operators encompass all of Ln, given an arbitraryGabor multiplierMg,γ

a on Rn, there exists a distribution σ ∈ S2n such that σ(X,D) =Mg,γ

a . By Theorem 1, this is equivalent, in terms of Schwartz kernels, to the equation

(9) T−F2σ =1

〈g, γ〉V ∗γ⊗gS

∗a,

which can be solved for σ in terms of a to yield

(10) σ =1

〈g, γ〉F−1

2 T −1− V ∗

γ⊗gS∗a.

In contrast, there is no obvious way to solve equation (9) for a in terms of σ, sinceneither of the operators V ∗

γ⊗g or S∗ is injective. Indeed, determining the Gaborsymbol that corresponds to a given Kohn-Nirenberg symbol seems to be a muchmore intricate problem.

2.3. Identical Gaussian windows. Let us specialize to the case of a window pairconsisting of identical Gaussians, g(t) = γ(t) = e−πt

2, where we use the notation

t2 = t · t for t ∈ Rn. The special structure of these windows means that we canobtain an alternate formula for the Schwartz kernel of any Gabor multiplier basedon them.

The key point in our proof of Theorem 1, at which we can take advantage of ourpresent choice of Gaussian windows, is the integral (8). The integrand involves theproduct

γ(τ − x)g(t− x) = e−π(τ−x)2e−π(t−x)2

= e−2π( τ+t2−x)2e−

π2(t−τ)2 .

Thus, in terms of the tempered change of variables

(v, w) = ( τ+t2, t− τ),

we see that γ(τ − x)g(t− x) has the form

(11) γ(τ − x)g(t− x) = G1(v − x)G2(w),



where G1(v) = e−2πv2 and G2(w) = e−π2w2

. Substituting (11) into the expression (8)for VgϕVγθ, we obtain

VgϕVγθ =

∫∫e2πi(t−τ)·ξG1(v − x)G2(w)θ⊗ϕ(τ, t) dτdt

=

∫∫e2πiw·ξG1(v − x)G2(w)θ⊗ϕ ◦ ψ−1(v, w)| det Jψ−1| dvdw

=

∫∫e2πiw·ξT(x,0)G1⊗G2(v, w)T ∗

ψ θ⊗ϕ(v, w) dvdw

=

∫e2πiw·ξ

(∫T(x,0)G1⊗G2(v, w)T ∗

ψ θ⊗ϕ(v, w) dv

)dw

= F−12

(G1⊗G2 ∗1 T ∗

ψ θ⊗ϕ),(12)

where G1(v) = G1(−v) and ∗1 denotes convolution with respect to the first n vari-ables. Replacing F−1

2 with F−1F1, and carrying F1 through the convolution, weobtain from (12) that

VgϕVγθ = F−1

(G1⊗G2F1T ∗

ψ θ⊗ϕ).

Proceeding as in the proof of Theorem 1, this leads to the following formula for theSchwartz kernel of Mg,γ

a :

(13) K(Mg,γa ) =

1

〈g, γ〉TψF−1

1 NbG1⊗G2Fa.

Recalling that g(t) = γ(t) = e−πt2, G1(v) = e−2πv2 and G2(w) = e−

π2w2

, we have

G1⊗G2(η, w) = 2−n2 e−

π2η2

e−π2w2

.

Since 〈g, γ〉 =∫e−2πt2 dt = 2n/2, the formula (13) simplifies as in the next proposition.

Proposition 5. Let g(t) = γ(t) = e−πt2

be identical Gaussians on Rn. Then for anya ∈ S ′

2n, the Schwartz kernel of Mg,γa is

(14) K(Mg,γa ) = TψF−1

1 NGFa,

where G : R2n → R is the Gaussian G(η, w) = e−π2(η,w)2 and ψ is the tempered change

of variables ψ(τ, t) = ( τ+t2, t− τ).

Each of the operators Tψ,F−11 , NG and F occurring in the expression (14) for the

Schwartz kernel of Mg,γa is injective. Therefore if Mg,γ

a = σ(X,D), we can, bycomparing the Schwartz kernels of the two operators, express the Gabor symbol ofMg,γ

a in terms of σ as

(15) a = F−1NHFσ, where H(η, y) = eπ2(η,y)2−πiη·y.

Moreover, for a given σ ∈ S ′2n, the operator σ(X,D) can be expressed as a Gabor

multiplier based on identical Gaussian windows only if the expression (15) yields a



well-defined tempered distribution a. (Note that (15) cannot be written as a convo-lution, because the rapidly increasing function H does not have a Fourier transform.)

Are there window pairs (g, γ) other than identical Gaussians for which one canobtain results analogous to these? The purpose of this paper is to describe a class ofwindow pairs for which this is possible.

3. Main results

3.1. Definition and examples of compatible window pairs. The expression(14) for the Schwartz kernel of a Gabor multiplier based on identical Gaussians is thebasis for the formula (15). With an eye to obtaining more general results along theselines, we investigate the class of all window pairs for which a formula analogous to(14) exists. In this regard the essential property of a Gaussian is the factorizationrule (11), so we focus on the class of all window pairs that obey such a rule. Thisleads directly to the following key technical notion.

Definition 6. We say that a window pair (g, γ) on Rn is compatible, and thatg and γ are compatible windows, if there exist functions G1, G2 : Rn → C and atempered change of variables (v, w) = ψ(τ, t) with w = t − τ such that for everyτ, t, x ∈ Rn,

γ(τ − x)g(t− x) = G1(v − x)G2(w).

Note that our derivation of formula (13) in Section 2.2 is valid for arbitrary compat-ible windows as we have just defined them. Hence we immediately have the followingresult.

Proposition 7. Let (g, γ) be a compatible window pair on Rn. Then for any a ∈ S ′2n,

the Schwartz kernel of Mg,γa is

K(Mg,γa ) =

1

〈g, γ〉TψF−1

1 NbG1⊗G2Fa,

where G1, G2 and ψ are related to (g, γ) as in Definition 6.

If (g, γ) is compatible then so is (γ, g). The precise connection between the ac-companying sets of functions as stated below may be easily verified. Here we writeψ↔ : R2n → R2n for the change of variables ψ↔(x, y) = (y, x), and we use the

notation F (x) = F (−x).

Proposition 8. Let (g, γ) be a compatible window pair on Rn, so that

γ(τ − x)g(t− x) = G1(v − x)G2(w),

in accordance with Definition 6. Then the reverse window pair (γ, g) is also compatibleand satisfies the equation

g(τ − x)γ(t− x) = G′1(v

′ − x)G′2(w

′),

where G′1 = G1, G

′2 = G2, and (v′, w′) = (v ◦ ψ↔, w).



Table 2 lists several families of compatible window pairs, alongside their corre-sponding factorizations and changes of variables, as prescribed by Definition 6. Theclass of compatible window pairs is easily seen to be invariant with respect to a num-ber of elementary operations: rescaling, translation and modulation of either window,and linear change of variables applied simultaneously to both windows. This obser-vation shows that the list in Table 2 is not complete. On the other hand, the windowpairs listed in Table 2 are reasonably representative. For example, it turns out thatevery non-negative compatible window pair on R1 can be obtained by translating,dilating, reversing or rescaling the given examples, or by reversing the order of awindow pair obtained this way. This is a rather involved technical result that will bepresented in a forthcoming paper [3].

Table 2. Examples of compatible windows. The parametersm,µ, α, k, κ appearing in lines (3.-5.) are positive scalars. The win-dow pairs of lines (1.-3.) are defined on Rn for any n; the window pairsof lines (4.,5.) are defined on R1 only.

Compatible window pairs (g, γ)

# g(t) γ(τ) (v, w) = ψ(τ, t) G1(v) G2(w)

1. 1 γ ∈ Sn (τ, t− τ) γ(v) 12. e2πiξ·t γ ∈ Sn (τ, t− τ) e2πiξ·vγ(v) e2πiξ·w

3. e−mt2 e−µτ2(

mt+µτm+µ , t− τ

)e−(m+µ)v2

e−mµ

m+µ w2

4. et−eαt

eτ−eατ(t+ 1

α log(

1+eα(τ−t)

2

), t− τ

)e2v−2eαv

(cosh(αw/2))−2/α

5. ekt−eαt

eκτ−eατ(t+ 1

α log(

1+eα(τ−t)

2

), t− τ

)e(k+κ)v−2eαv

(e

καk+κ w + e−

kαk+κ w

2

)− k+κα

The example of line (1.), Table 2, which satisfies the definition of compatibility ina trivial way, is not directly of practical interest. Of course the Gaussian windows ofline (3.) are the prototype for the notion of compatibility, and they are a standardwindow choice for the short-time Fourier transform. The most interesting examplesin Table 2 are the windows of line (4.) and their generalizations in line (5.). Thesewere discovered using the alternate characterization of compatibility developed inthe next section of this paper; their full derivation will be presented in [3]. LikeGaussians, these windows have a fundamental role in probability theory and arise inconnection to a variant of the central limit theorem. More precisely, let E denote thefunction g of line (4.) in the special case α = 1:

E(t) = et−et

.

Properly scaled affine transformations of E, of the form

1

bE(a−t

b),

are the density functions of so-called “extreme value” probability distributions, de-scribed by Fisher and Tippett in [2]. Based on this, we refer to the windows of lines(4.,5.) as extreme value windows—see Figure 1.



Figure 1. Extreme value windows. The function f(t) = et−eαt

, above,and its reversal f(−t), below, are plotted for α = 10. Note the f(−t)is close to being causal, i.e. zero for negative values of t. This isbecause the double exponential e−e

−αttends to the Heaviside function

with increasing α.

Note that the Fourier transform of an extreme value window is proportional to thegamma function, restricted to a line of constant real part:

F(ekt−e

αt)

(ξ) =1

αΓ

(k

α− 2πi

αξ

).

Thus the Fourier transform of an extreme value distribution is analytic, never zero,and rapidly decreasing. In Section 3.4, we show that extreme value windows have acertain advantage over Gaussians with respect to the representation of linear opera-tors.

3.2. An alternate characterization of compatibility. Here we develop an al-ternate characterization of compatibility that makes it easier to derive formulas andto prove some technical facts that will be needed in subsequent sections. The argu-ments given here are purely technical; they build toward our main results which arepresented in the following Section 3.3.

Let (g, γ) be a compatible pair on Rn with corresponding functions G1, G2 andtempered change of variables (v, w) = ψ(τ, t), as stipulated in Definition 6. It can bededuced from the basic equation,

(16) γ(τ − x)g(t− x) = G1(v − x)G2(w) ∀τ, t, x ∈ Rn,

that there are several simplifying assumptions to be made concerning G1, G2 and ψwithout any incumbent loss of generality.



Firstly, note that

G1(v(τ, t)− x)G2(t− τ) = γ(τ − x)g(t− x)

= γ((τ − t)− (x− t))g(0− (x− t))

= G1(v(τ − t, 0)− (x− t))G2(t− τ)

= G1(v(τ − t, 0) + t− x)G2(w).(17)

This leads to the first of our simplifying propositions.

Proposition 9. It may be assumed without loss of generality that

(18) v(τ, t) = v(τ − t, 0) + t ∀τ, t ∈ Rn.

Proof. Suppose that (18) fails to hold, and consider the new function v′, defined by

v′(τ, t) = v(τ − t, 0) + t ∀τ, t ∈ Rn.

Based on the fact that ψ = (v, w) is a tempered change of variables, it may be verifiedthat the change of variables ψ′ = (v′, w) is tempered also. And by construction,

v′(τ, t) = v′(τ − t, 0) + t ∀τ, t ∈ Rn.

Now, applying the identity (17) yields

G1(v − x)G2(w) = G1(v(τ − t, 0) + t− x)G2(w)

= G1(v′ − x)G2(w).

Therefore the defining equation (16) remains satisfied if we replace ψ by the newchange of variables ψ′.

Proposition 10. It may be assumed without loss of generality that v(0, 0) = 0.

Proof. The equation that defines compatibility, (16), remains satisfied if G1 isreplaced by

G′1(y) = G1(y + v(0, 0))

and v is replaced byv′(τ, t) = v(τ, t)− v(0, 0).

The modified change of variables ψ′ = (v′, w) is tempered, and by constructionv′(0, 0) = 0.

Proposition 11. Without loss of generality, G2(0) = 1 and G1 = gγ.

Proof. Substituting w = 0 (i.e., t = τ) and x = 0 into the equation (16) yields

(19) γ(t)g(t) = G1(v(t, t))G2(0).

Since 0 6= 〈g, γ〉 =∫gγ, it follows from (19) that that G2(0) 6= 0. The equation

(16) evidently remains satisfied if G2 is replaced by 1G2(0)

G2 and G1 is replaced by

G2(0)G1, so we may assume G2(0) = 1.Assuming in addition that (18) holds and that v(0, 0) = 0, we have that v(t, t) =

v(0, 0) + t = t. It then follows from equation (19) that G1(t) = γ(t)g(t).It will streamline subsequent arguments to assume that G1, G2 and ψ have the

forms given in Propositions 9, 10 and 11. We now introduce the notion of “product-translation commutativity” for a window pair.



Definition 12. Let (g, γ) be a window pair on Rn. We say that (g, γ) is product-translation commutative, or that g and γ are product-translation commutative,if there exist functions C : Rn → C and ζ : Rn → Rn such that for every w ∈ Rn,

(20) gTwγ = C(w)Tζ(w)(gγ),

and such that (τ, t) 7→ (t− ζ(t− τ), t− τ) is a tempered change of variables.

Although it is not immediately obvious, product-translation commutativity of awindow pair is equivalent to compatibility. This fact facilitates the analysis of com-patibility.

Theorem 2. A window pair (g, γ) is product-translation commutative if and only ifit is compatible. Moreover, if a window pair (g, γ) has this property then, without lossof generality, the terms occurring in the respective defining equations,

gTwγ = C(w)Tζ(w)(gγ) and γ(τ − x)g(t− x) = G1(v − x)G2(w),

may be assumed to be related by the formulas

G2 = C, v(τ, t) = t− ζ(t− τ) and ζ(y) = −v(−y, 0).

Proof. We prove the equivalence of Definitions 6 and 12; the stated formulas emergein the course of the proof.

Suppose that (g, γ) is compatible, with associated functions G1, G2 and ψ thatsatisfy the properties in Propositions 9, 10 and 11. The basic equation of Definition 6,equation (16), can be written symbolically as

T(x,x)γ ⊗ g = TψT(x,0)G1 ⊗G2

⇔ γ ⊗ g = T(−x,−x)TψT(x,0)G1 ⊗G2

⇔ γ ⊗ g(τ, t) = G1(v(τ + x, t+ x)− x)G2(w).(21)

Define ζ(y) = −v(−y, 0), so that substituting x = −t into (21) yields

γ ⊗ g(τ, t) = G1(v(τ − t, 0) + t)G2(w)

= G1(t− ζ(w))G2(w)

= g(t− ζ(w))γ(t− ζ(w))G2(w) (by Proposition 11).(22)

The identity w = t−τ implies that γ⊗g(τ, t) = gTwγ(t). Therefore (22) is equivalentto the equation

(23) gTwγ(t) = G2(w)Tζ(w)(gγ)(t).

Note that v(τ, t) = v(τ − t, 0) + t = t− ζ(t− τ), so the tempered change of variablesψ = (v, w) is precisely the map (τ, t) 7→ (t − ζ(t − τ), t − τ). Together with (23),this verifies that the pair (g, γ) conforms to Definition 12 of product-translationcommutativity, with C = G2.

Conversely, suppose that the window pair (g, γ) is product-translation commuta-tive, with associated functions C, ζ as in Definition 12. Set v(τ, t) = t − ζ(t − τ),



so that (v(τ, t), w(τ, t)) = (t− ζ(t− τ), t− τ) is a tempered change of variables. Asbefore, γ ⊗ g(τ, t) = gTwγ(t), so by definition,

γ ⊗ g(τ, t) = γ(t− ζ(w))g(t− ζ(w))C(w)

⇒ T(x,x)γ ⊗ g(τ, t) = γ(t− ζ(w)− x)g(t− ζ(w)− x)C(w)

= (γg)(v − x)C(w).

Thus (g, γ) conforms to Definition 6, with associated functions G1 = gγ, G2 = C andψ = (v, w).

We are now in a position to derive two additional formulas concerning compatiblewindow pairs. Recall that X : Rn → Rn denotes the identity map; Xk denotesits k-th component. We write (gX) ∗ ˜γ for the vector

((gX1) ∗ ˜γ, . . . , (gXn) ∗ ˜γ),

and similarly 〈gX, γ〉 = (〈gX1, γ〉, . . . , 〈gXn, γ〉). A tilde over a function indicates

reflection in the argument: F (y) = F (−y).

Proposition 13. If (g, γ) is a compatible pair then, without loss of generality, theassociated function G2 is given by the formula

(24) G2 =g ∗ ˜γ〈g, γ〉

.

And the change of variables ψ can be expressed as ψ(τ, t) = (t−ζ(t−τ), t−τ), wherethe restriction of ζ to points w at which G2(w) 6= 0 is given by the formula

(25) ζ =(gX) ∗ ˜γg ∗ ˜γ − 〈gX, γ〉

〈g, γ〉.

Proof. By Theorem 2, g and γ satisfy the equation

gTwγ = G2(w)Tζ(w)(gγ).

Integrating both sides over Rn and rearranging terms yields

G2(w) =

∫gTwγ∫

Tζ(w)(gγ)

=g ∗ ˜γ(w)∫

gγ

=g ∗ ˜γ(w)

〈g, γ〉.

Observe that

(gX) ∗ ˜γ(w) =

∫XgTwγ

=

∫XG2(w)Tζ(w)(gγ)

= G2(w)

∫(X + ζ(w))gγ.

Also,g ∗ ˜γ = G2(w)〈g, γ〉,



so, provided G2(w) 6= 0, the right-hand side of (25) becomes

G2(w)∫

(X + ζ(w))gγ

G2(w)〈g, γ〉− 〈gX, γ〉

〈g, γ〉= ζ(w)

〈g, γ〉〈g, γ〉

+〈gX, γ〉〈g, γ〉

− 〈gX, γ〉〈g, γ〉

= ζ(w).

It turns out that G2 can never be zero, so that in fact the formula (25) is valideverywhere, but we will not prove this in the present paper (see [3]).

Next we prove some technical results, in anticipation of the coming section, wherewe relate the Gabor symbols of Gabor multipliers based on compatible windows totheir Kohn-Nirenberg symbols.

Proposition 14. Let (g, γ) be a compatible window pair, with associated functionsG1, G2. Then without loss of generality

1

〈g, γ〉G1 ⊗G2 =

1

|〈g, γ〉|2(ˆg ∗ γ)⊗ (g ∗ γ).

Proof. This is just an application of Propositions 11 and 13.Recall that ψ− denotes the particular change of variables ψ−(τ, t) = (τ, t− τ) and

that T− denotes the corresponding operator Tψ− .

Lemma 15. Let (g, γ) be a compatible window pair, with associated change of vari-ables ψ(τ, t) = (t− ζ(t− τ), t− τ) (as in Theorem 2). Then

ψ− ◦ ψ−1(x, y) = (x+ ζ(y)− y, y).

Proof. Write (f1(x, y), f2(x, y)) = ψ−1(x, y). Then

(x, y) = ψ ◦ ψ−1(x, y) = (f2 − ζ(f2 − f1), f2 − f1).

Combining the above expressions for x and y yields that

f1 = x− y + ζ(y).

Thus,ψ− ◦ ψ−1(x, y) = ψ−(f1, f2) = (f1, f2 − f1) = (x+ ζ(y)− y, y).

Proposition 16. Let ψ(τ, t) = (t−ζ(t−τ), t−τ) be the change of variables associatedto a compatible window pair (g, γ). Then F1Tψ−1T− = e−2πiϕF1, where

ϕ(η, y) = η · (y − ζ(y)).

Proof. First note that Tψ−1T− = Tψ−◦ψ. Let F (τ, y) ∈ S2n be a dummy function,with τ serving as Fourier dual variable to η. Then(

F1Tψ−◦ψF)(η, y) =

∫e−2πiτ ·ηTψ−◦ψF (τ, y) dτ

=

∫e−2πiτ ·ηF (τ + ζ(y)− y, y) dτ (by Lemma 15)

= e−2πiη·(y−ζ(y))∫e−2πiτ ·ηF (τ, y) dτ

= e−2πiη·(y−ζ(y))F1F (η, y).



By density of S2n in S ′2n and continuity of the operators in question, it follows that

F1Tψ−1T− = e−2πiϕF1.

3.3. Relation between the Gabor and Kohn-Nirenberg symbols. In Sec-tion 2.3 we derived an explicit formula for the Gabor symbol of a Gaussian Gabormultiplier in terms of its Kohn-Nirenberg symbol, and then used this to constrain theclass of Kohn-Nirenberg operators that can be realized as such a Gabor multiplier.We now present the corresponding results in a much more general setting, namely,for arbitrary compatible windows.

Theorem 3. Let a ∈ S ′2n and let (g, γ) be a compatible window pair on Rn. Then

Mg,γa = σ(X,D) if and only if the symbols a and σ are related by the equation

(26)1

|〈g, γ〉|2(ˆg ∗ γ)⊗ (g ∗ γ)Fa = e−2πiϕFσ,

where the phase function ϕ is given in terms of (g, γ) by the formula

ϕ(η, y) = η ·(y −

((gX) ∗ ˜γ(y)g ∗ ˜γ(y) − 〈gX, γ〉

〈g, γ〉

)).

Proof. Comparing the Schwartz kernel of Mg,γa , given in Proposition 7, to the

Schwartz kernel of σ(X,D), given in (5), we have

1

〈g, γ〉TψF−1

1 NbG1⊗G2Fa = T−F2σ

⇐⇒ 1

〈g, γ〉G1 ⊗G2Fa = F1Tψ−1T−F2σ.(27)

Applying Proposition 14 to the left-hand side of (27), and Proposition 16 togetherwith the formula (25) of Proposition 13 to the right-hand side, yields the desiredequation.

It is perhaps clearer to formulate equation (26) with the terms that depend on thewindows combined into a single function F , as follows:

(28) FFa = Fσ, where F =e2πiϕ

|〈g, γ〉|2(ˆg ∗ γ)⊗ (g ∗ γ).

Of course one can solve explicitly for the Gabor symbol to yield

(29) a = F−1N1/FFσ,with F as in (28). For the case where g and γ are compatible, equation (29) consti-tutes the companion result to equation (10) in Section 2.2.

The expressions

1

|〈g, γ〉|2(ˆg ∗ γ)⊗ (g ∗ γ) and e−2πiϕ

corresponding to each of the compatible pairs in Table 2 are worked out in Ta-ble 3, where the consequent relations between Gabor and Kohn-Nirenberg symbols



are listed. The following characterization of operators that can be represented as a

Table 3. Comparison of Gabor and Kohn-Nirenberg symbols for com-patible windows. Note that, as in Table 2, m,µ, α, k, κ denote positivescalars, and the windows of lines (4.,5.) are defined on R1 only.

Gabor vs. Kohn-Nirenberg symbols for compatible window pairs (g, γ)

# g(t) γ(τ) Relation between Fa(η, y) and Fσ(η, y)

1. 1 γ ∈ Sn1bγ(0)γ(η)Fa = Fσ

2. e2πiξ·t γ ∈ Sn1bγ(−ξ)

γ(η + ξ)e−2πiξ·yFa = Fσ

3. e−mt2 e−µτ2e−

π2m+µ η2

e−mµ

m+µ y2Fa = e−

2πmm+µ iη·yFσ

4. et−eαt

eτ−eατ 22/α

Γ(2/α)Γ(

2α −

2πiα η)

ey

(1+eαy)2/αFa = (1 + eαy)−2πiη

α Fσ

5. ekt−eαt

eκτ−eατ 1

Γ( k+κα )Γ

(k+κ

α − 2πiα η)(e κα

k+κ y + e−kα

k+κ y

2

)− k+κα

Fa = (1 + eαy)−2πiη

α Fσ

Gabor multiplier based on a given window pair rests essentially on Theorem 3.

Theorem 4. Let σ ∈ S ′2n. The Kohn-Nirenberg operator σ(X,D) can be represented

as a Gabor multiplier based on a given compatible window pair (g, γ) if and only ifσ ∈ GS ′

2n, or equivalently, σ ∈ H ∗ S ′2n, where

G = (ˆg ∗ γ)⊗ (g ∗ γ), H = (gγ)⊗ (¯gγ).

Proof. Since the change of variables (τ, t) 7→ (t− ζ(t− τ), t) associated to (g, γ) isrequired to be tempered, the function

h(η, y) = e−2πiϕ(η,y) = e−2πiη·(y−ζ(y))

has the property that

hS ′2n = (1/h)S ′

2n = S ′2n.

Now, for fixed σ ∈ S ′2n, the equation (26) of Theorem 3 holds for some a ∈ S ′

2n if andonly if

Fσ ∈ (ˆg ∗ γ)⊗ (g ∗ γ)|〈g, γ〉|2 h

S ′2n = (ˆg ∗ γ)⊗(g ∗ γ)S ′

2n ⇐⇒ σ ∈((gγ)⊗ (¯gγ)

)∗ S ′

2n.

We can read off from this result some basic heuristic principles concerning theexistence of a Gabor multiplier based on given windows that represents a given Kohn-Nirenberg operator, as follows. In order to represent σ(X,D) as a Gabor multiplierbased on the compatible window pair (g, γ), it has to be the case that σ(η, y) decaysas fast as ˆg ∗ γ in the variables η, and as fast as g ∗ γ in the variables y. The symbolσ(x, ξ) itself has to be as smooth as gγ in the variables x, and as smooth as ¯gγ in thevariables ξ.



Theorem 4 shows that for Gabor multipliers based on a given window pair (g, γ)to represent a wide class of linear operators, the function H should approximate adelta function, so that H ∗S ′

2n∼= S ′

2n. Note that the scope of identical windows g = γis limited by the uncertainty principle: it is not possible for gγ and ¯gγ to both behighly localized, so the symbol σ(x, ξ) of an operator representable using identicalwindows must be slowly varying in either x or ξ. This constraint can be avoided byusing distinct windows g, γ, provided one of g, γ is highly localized while the Fouriertransform of the other is highly localized. In keeping with this assertion, we will seein Section 3.5 that maximum generality is achieved with the “singular” window pair(g, γ) = (1, δ) and its reversal (δ, 1).

3.4. Generality of Gaussian versus extreme value windows. With respect tothe realization of linear operators as Gabor multipliers, the generality of a givenwindow pair (g, γ) is represented by the set

R(g, γ) = {σ ∈ S ′2n | ∃a ∈ S ′

2n such that σ(X,D) = Mg,γa } .

And so, given two window pairs (g1, γ1), (g2, γ2) on Rn, we say that (g1, γ1) is moregeneral than (g2, γ2) if

R(g2, γ2) ⊆ R(g1, γ1),

or equivalently, if

R(g2, γ2) ⊆ R(g1, γ1).

In terms to this notation, Theorem 4 says that for a compatible window pair on Rn,

R(g, γ) = (gγ)⊗ (¯gγ) ∗ S ′2n and R(g, γ) = (ˆg ∗ γ)⊗ (g ∗ γ)S ′

2n,

which allows us to compare the generality of any two compatible window pairs. Forexample, the next result shows that extreme value windows are more general thanGaussians.

Proposition 17. Let α,m, µ > 0 be fixed. On R1, the window pair

(g1(t), γ1(t)) =(et−e

αt

, et−eαt)

is more general than the window pair

(g2(t), γ2(t)) =(e−mt

2

, e−µt2).

Proof. We carry out some computations so as to be able to apply Theorem 4. Write

K1 = ( ˆg1 ∗ γ1)⊗ (g1 ∗ γ1),

K2 = ( ˆg2 ∗ γ2)⊗ (g2 ∗ γ2).

From Table 3,

1|〈g1,γ1〉|2

K1(η, y) = 22α +

2πiηα

Γ(2/α)Γ(

2α− 2πi

αη)

ey

(1+eαy)2/α ,

1|〈g2,γ2〉|2

K2(η, y) = e−π2

m+µη2

e−mµ

m+µy2 .



Note that K1S ′2 = K ′

1S ′2, where

K ′1(η, y) =

(2

2α +

2πiηα

Γ(2/α)

)−11

|〈g1,γ1〉|2K1(η, y)

= Γ(

2α− 2πi

αη)

ey

(1+eαy)2/α .

And K2S ′2 = K ′

2S ′2, where

K ′2(η, y) =

1

|〈g2, γ2〉|2K2(η, y)

= e−π2

m+µη2

e−mµ

m+µy2 .

We claim that K ′2/K

′1 ∈ P2, from which it follows that (K ′

2/K′1)S ′

2 ⊆ S ′2. To verify

the claim, it suffices to compute the rate of decay of K ′1 and to see that it is less than

that of K ′2. By Stirling’s approximation this works out to be

|K ′1(η, y)| ∼ |η|

2α− 1

2 e−(π2

α|η|+|y|),

which is slower than the Gaussian decay of K ′2.

Now, let ρ ∈ S ′2 be arbitrary, and consider the distribution K ′

2ρ:

K ′2ρ = K ′

1 ((K ′2/K

′1)ρ) ∈ K ′

1S ′2,

which proves that K ′2S ′

2 ⊆ K ′1S ′

2, or equivalently K2S ′2 ⊆ K1S ′

2. By Theorem 4, thisshows that (g1, γ1) is more general than (g2, γ2).

3.5. Singular windows. The class of window pairs established in Definition 2 isuseful because Gabor multipliers based on them are well-defined irrespective of theGabor symbol. We were thus free to carry out a somewhat general analysis, unen-cumbered by questions of interpretation or scope of validity of formulas. But if oneis interested, not in a general analysis, but rather in studying particular operatorssuch as, say, a specific type of pseudodifferential operator, then there is good reasonto consider windows of a more general type. This comes of course at a price—Gabormultipliers based on more general windows may be well-defined only for a limitedclass of Gabor symbols, and one has to carry out some sort of ad hoc analysis todetermine this class. We will consider a few particular examples of such “singular”windows. To begin, we establish the relevant definitions.

Definition 18. If g, γ ∈ S ′n are such that one of the expressions 〈g, γ〉 or 〈γ, g〉 is

well-defined and non-zero, but (g, γ) is not a window pair in the sense of Definition 2,then we call (g, γ) a singular window pair on Rn.

Definition 19. We say that a singular window pair (g, γ) on Rn is compatible,and that g and γ are singular compatible windows, if there exist distributionsG1, G2 ∈ S ′

n and a change of variables (v, w) = ψ(τ, t) (not necessarily tempered)with w = t− τ such that for every x ∈ Rn, the equation

T(x,x)γ ⊗ g = TψT(x,0)G1 ⊗G2

holds in the sense of distributions.



Perhaps the simplest way to obtain singular compatible windows is as limits ofregular compatible windows. For instance, if γk → δ is an approximate identity,then the compatible pairs (1, γk), which have a consistent change of variables, tendto the limit (1, δ). The pair (1, δ) is indeed singular compatible, with accompanyingfunctions G1 = δ,G2 = 1 and change of variables ψ(τ, t) = (τ, t− τ). Note that, forevery ϕ, θ ∈ Sn,

V1ϕ(x, ξ) = ϕ(ξ), Vδθ(x, ξ) = e−2πix·ξθ(x).

Thus

(30) 〈M1,δa ϕ, θ〉 = 〈a, e−2πix·ξθ ⊗ ϕ〉.

Since e−2πix·ξθ⊗ ϕ ∈ S2n, the equation (30) shows that the Gabor multiplier M1,δa is

well-defined for every a ∈ S ′2n.

The Schwartz kernel of M1,δa may be computed directly from (30):

e−2πix·ξθ ⊗ ϕ = e−2πix·ξF−12 θ ⊗ ϕ

= F−12 T ∗

−θ ⊗ ϕ

= (T−F2)∗ θ ⊗ ϕ.

It follows that

(31) K(M1,δa ) = T−F2a.

(We could have alternatively applied the formula of Proposition 7 to obtain the sameresult.) The expression (31) is of course the Schwartz kernel of the Kohn-Nirenbergpseudodifferential operator having symbol a. That is, changing notation slightly sothat σ denotes the Gabor symbol in place of a, we have:

Proposition 20. For every σ ∈ S ′2n, M1,δ

σ = σ(X,D).

This result is of theoretical importance in that it shows Gabor multipliers to becompletely general, at least once we allow singular windows. Applying Proposition 8and the adjoint formula of Proposition 4 yields that

Mδ,1a =

(M1,δ

a

)∗for any a ∈ S ′

2n. Thus (δ, 1) too is a singular compatible pair which gives rise to awell-defined Gabor multiplier irrespective of the symbol. Although they are genuineexamples of singular compatible windows, the pairs (1, δ) and (δ, 1) do not reallygive anything new, since Gabor multipliers based on them are just Kohn-Nirenbergoperators (or their adjoints) in disguise.

On the other hand, if we let α tend to ∞ in the compatible pair(ekt−e

αt

, eκτ−eατ)

on R1, we obtain something new, namely the pair

(32) (g(t), γ(τ)) =(H(−t)ekt, H(−τ)eκτ

),



where H denotes the Heaviside jump function and k, κ > 0 are constants. Theassociated functions G1(v) = H(−v)e(k+κ)v,

G2(w) = e(kH(−w)−κH(w))w =

{ekw w ≤ 0e−κw w > 0

,

and the change of variables (v, w) = ψ(τ, t) = (max{τ, t}, t − τ), verify that (32)conforms to the definition of a singular compatible window pair. By Proposition 7the Gabor multiplier Mg,γ

a based on the windows (32) has kernel

(33) K(Mg,γa ) =

1

〈g, γ〉TψF−1

1 NbG1⊗G2Fa,

provided the latter is well-defined. A short calculation yields that

1

〈g, γ〉G1 ⊗G2(η, y) =

k + κ

k + κ− 2πiηe(kH(−y)−κH(y))y.

Note that e(kH(−y)−κH(y))y is not differentiable at y = 0, which implies that if a(η, y)is singular at some point (η, 0) then the usual distributional calculus breaks down forthe right-hand side of (33). In this case the interpretation of Mg,γ

a requires additionalclarification (which in practice can be quite straightforward).

We have already pointed out that for any γ ∈ Sn, the pair (1, γ) is compatible,and hence so is its reversal (γ, 1). In dimension n = 1, a limiting case of the latter isthe pair

(34) (γ, 1), γ(t) = H(−t)et.

The truncated exponential γ in the pair (34) is advantageous for a couple of reasons.Not only is it amenable to explicit calculation, but also the singularity at 0 rendersthe pair (γ, 1) more general than if γ were a Schwartz class function, as we will nowshow. Note first that (34) is singular compatible with associated functions

G1 = γ,G2 = 1, ψ(τ, t) = (t, t− τ).

And the Gabor multiplier Mγ,1a makes sense for arbitrary symbol a ∈ S ′

2. Further-more,

1

〈γ, 1〉G1 ⊗G2(η, y) =

1

1− 2πiη, F1Tψ−1T−F2 = e−2πiϕF ,

where ϕ(η, y) = ηy. Thus if a Kohn-Nirenberg operator σ(X,D) can be representedas a Gabor multiplier Mγ,1

a based on the windows (34), then by (27) the Gabor andKohn-Nirenberg symbols are related by the equation

1

1− 2πiηFa(η, y) = e−2πiηyFσ(η, y)

⇐⇒ a(x, ξ) =((1− 2πiξ)e2πixξ

)∗ σ.(35)

The equation (35) imposes no restrictions on σ for the existence of a correspondingGabor symbol a. In other words, the window pair (34) is completely general:

R(H(−t)et, 1) = S ′2.



4. Summary

The starting point for the present paper was the two-fold premise that (i) in practiceone often encounters linear operators in Kohn-Nirenberg form, σ(X,D), and that (ii)it is of interest both from the numerical and analytical points of view to expresssuch an operator as a Gabor multiplier Mg,γ

a = V ∗γ NaVg, for some choice of localizing

windows g, γ. We have determined a class of window pairs, namely, compatible pairsand singular compatible pairs, for which there exists an explicit formula by which tocompute the Gabor symbol a in terms of the given Kohn-Nirenberg symbol σ. Moreprecisely, the ratio σ/a of the Fourier transforms of the symbols is given in terms ofthe windows by the expression

(36)e2πiϕ

|〈g, γ〉|2(ˆg ∗ γ)⊗ (g ∗ γ),

where the phase function ϕ is

(37) ϕ(η, y) = η ·(y −

((gX) ∗ ˜γ(y)g ∗ ˜γ(y) − 〈gX, γ〉

〈g, γ〉

)).

Extreme value windows are a particularly interesting example of compatible windowswhich are approximately causal (or reverse causal), and are more general than Gaus-sians with respect to the representation of linear operators. The class of singular com-patible windows includes window pairs in terms of which every linear operator maybe represented as a Gabor multiplier. Within this framework, the Kohn-Nirenbergformulation itself is an example of a Gabor multiplier, based on the singular windowpair (1, δ). These results offer the prospect of investigating (or evaluating) particularlinear operators, for example partial differential operators or nonstationary filters, interms of their myriad representations as Gabor multipliers.

In conclusion we mention a couple of open problems stemming from the presentpaper. Concerning the formula (36) for the ratio of the transformed symbols, it isknown to be valid for compatible and singular compatible windows. But are thereother, non-compatible windows for which it is also valid? More generally, is there abroader class of window pairs, for which formulas analogous to (36) and (37) can beobtained?

References

1. Hans G. Feichtinger and Krysztof Nowak, A first survey of Gabor multipliers, in Advances inGabor analysis, Hans G. Feichtinger and Thomas Strohmer, editors, Applied and NumericalHarmonic Analysis, Birkhauser, Boston, 2003.

2. R. A. Fisher and L. H. C. Tippett, Limiting forms of the frequency distribution of the largest orsmallest member of a sample, Proceedings of the Cambridge Philosophical Society, XXIV Pt. 2(1928), 180-190.

3. Peter C. Gibson and Petr Zizler, Compatible windows in Gabor analysis (working title), inpreparation.

4. Karlheinz Grochenig, Foundations of time-frequency analysis, Applied and Numerical HarmonicAnalysis, Birkhauser, Boston, 2001.



5. Michael P. Lamoureux, Peter C. Gibson, Jeff P. Grossman and Gary F. Margrave, A fast,discrete Gabor transform via a partition of unity, 35pp., submitted to The Journal of FourierAnalysis and Applications.

6. Gary F. Margrave and Robert J. Ferguson, Wavefield extrapolation by nonstationary phaseshift, Geophysics, 64 (1999), 1067-1078.

7. Gary F. Margrave, Robert J. Ferguson, and Michael P. Lamoureux, 2002, Approximate FourierIntegral Wavefield Extrapolators for Heterogeneous, Anisotropic Media, Canadian AppliedMathematics Quarterly, in press.

8. Gary F. Margrave, Michael P. Lamoureux, Jeff P. Grossman, and V. Iliescu, Gabor deconvolu-tion of seismic data for source waveform and Q correction, in 72nd Ann. Internat. Mtg, Soc. ofExpl. Geophys., Expanded Abstract Volume, (2002), 2190-2193.

9. Richard Rochberg and Kazuya Tachizawa, Pseudodifferential operators, Gabor frames, andlocal trigonometric bases, in Gabor analysis and algorithms: Theory and applications, HansG. Feichtinger and Thomas Strohmer, editors, Applied and Numerical Harmonic Analysis,Birkhauser, Boston, 1998.

10. Laurent Schwartz, Theorie des noyaux, Proceedings of the International Congress of Mathemati-cians 1 (1950), 220–230.

Peter C. Gibson, Department of Mathematics & Statistics, University of Calgary,Calgary, Alberta T2N 1N4


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