Elena Cordero

Time–Frequency analysis of localization operators

Abstract We study a class of pseudodifferential operators known as time–frequency localization operators, Anti-Wick operators, Gabor–Toeplitz operators or wave packets. Given a symbol a and two windows ϕ1,ϕ2, we investigate the multilinear mapping from (a,ϕ 1 ,ϕ 2 )∈ S ′( R 2d )× S ( R d )× S ( R d ) to the localization operator Aaϕ1,ϕ2 and we give sufficient and necessary conditions for Aaϕ1,ϕ2 to be bounded or to belong to a Schatten class. Our results are formulated in terms of time–frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.

Banach Gelfand Triples for Gabor Analysis

It is the purpose of this survey note to show the relevance of a Gelfand triple which is closely connected with time–frequency analysis and Gabor analysis. The Segal algebra S 0(ℝ d ) and its dual can be shown to be — for a large variety of concrete cases – a convenient substitute for the Schwartz space S(ℝ d ) and it’s dual, the space of tempered distributions S′(ℝ d ). This concrete pair of Banach spaces is actually a Gelfand triple, which allows to describe in a very intuitive way the properties of the classical Fourier transform and other unitary operators arising in the treatment of various mathematical questions, e.g., multipliers in harmonic analysis. We will demonstrate the usefulness of the Banach Gelfand triple (S 0(ℝ d ), L 2(ℝ d ), S 0(ℝ d )) within time–frequency analysis, with a special emphasis on questions from time–frequency analysis and Gabor analysis.

Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential operators in a Sjostrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sjostrand class consists of generalized metaplectic operators. As a consequence, the Schrodinger equation preserves the phase-space concentration, as measured by modulation space norms.

Wick calculus: A time-frequency approach

The purpose of this paper is to present a formula for the product of two Wick operators, defined in terms of different pairs of windows φ1, φ2. In principle, Wick operators can be converted to Weyl operators, and hence one may apply to them the standard symbolic calculus [14, 21]. It is natural, however, to consider the product in the Wick form, and try to compute directly the symbol in terms of the symbols of the factors; see in this direction [3, 5, 15, 16, 19]. Recently Ando and Morimoto [1] have given a full expansion for the Wick symbol of the product in the case when all the windows coincide with the Gaussian function. We propose here a general formula. The expression is somewhat non-standard, because we write the product as a sum of anti-Wick operators corresponding to a sequence of different pairs of windows, with decreasing order. This seems to us the only possible expression of reasonable simplicity in the generic case. In the remaining part of this Introduction we recall the definition of Wick operators and state the composition result. In Section 2 we summarize some concepts of time-frequency methods used in the proof. In Section 3 we introduce the classes of symbols we are arguing on. They are, essentially, those of Shubin [18], as generalized in [3]. Let us emphasize that other classes of symbols, under weaker assumptions on derivative estimates, would work as well. In Section 4 we prove the result. In Section 5 we give a composition formula for the particular case of Gaussian functions as a pair of windows and we recapture the results of Lerner [15, 16], Ando and Morimoto [1]. Section 6 is devoted to miscellaneous comments. Namely we show how to pass from a pair of windows to another and, finally, we construct a parametrix for the elliptic Wick operators by using our formula; a natural application, which we hope to detail in future papers, concerns regularity results in the frame of the modulation spaces [9, 12]. Before stating the precise definition, let us observe that Wick operators have been considered in the past under rather different points of view, and different names. They were introduced by Berezin [2] as a quantization procedure, and as an approximation of pseudodifferential operators (“wave packets”) by Cordoba and Fefferman [7, 11]. From the point of view of the time-frequency analysis, which we shall adopt in the following, they have been studied by Daubechies [8] and

Necessary conditions for schatten class localization operators

We study time-frequency localization operators of the form A φ1,φ2 α , where a is the symbol of the operator and φ1, φ2 are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for A φ1,φ2 α ∈ S p (L 2 (R d )), the Schatten class of order p, is that a belongs to the modulation space M p, ∞(R 2d ) and the window functions to the modulation space M 1 . Here we prove a partial converse: if A φ1,φ2 α ∈ S p (L 2 (R d )) for every pair of window functions φ 1 , φ 2 ∈ S(R 2d ) with a uniform norm estimate, then the corresponding symbol a must belong to the modulation space M p, ∞(R 2d ). In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p = ∞ and p = 2, we recapture earlier results, which were obtained by different methods.

Gabor representations of evolution operators

We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodierential operators with symbols of Gevrey, analytic and ultra- analytic type. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrodinger-type propagators (14), surprisingly reveal to be an equally ecient tool for representing solutions to hyperbolic and parabolic-type dierential equations with constant coecients. In fact, the Gabor matrix represen- tation of the corresponding propagator displays super-exponential decay away from the diagonal.

Wave packet analysis of Schrödinger equations in analytic function spaces

Abstract We consider a class of linear Schrodinger equations in R d , with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic singularities.

Gelfand-Shilov Window Classes for Weighted Modulation Spaces

We show that the Gelfand–Shilov algebra is densely embedded in the weighted modulation space . Here the weight function m is allowed to have a super-exponential growth at infinity. The basic tool is given by an integral transform called short-time Fourier transform (STFT). The STFT is used to both define and characterize the previous spaces. Moreover, our result is attained using the properties of the STFT and its adjoint.

Anti-Wick quantization with symbols in ^{} spaces

We give a classification of pseudo-differential operators with anti-Wick symbols belonging to L p spaces: if p = 1 the corresponding operator belongs to trace classes; if 1 < p < 2 we get Hilbert-Schmidt operators; finally, if p < oo, the operator is compact. This classification cannot be improved, as shown by some examples.

Microlocal Analysis and Applications

We first give a short survey on the methods of Microlocal Analysis. In particular we recall some basic facts concerning the theory of pseudodifferential operators. We then present two applications. We first discuss lower bounds for operators with multiple characteristics. Then we give a new formula for the composition of Wick operators.