Catálogo de publicaciones - libros

The concept of star discontinuity is defined for functions of several variables. A star discontinuity in dimension one is simply a jump discontinuity. It is then shown that in arbitrary dimensions the Gibbs phenomenon for square convergence occurs for periodic functions satisfying appropriate hypotheses at star discontinuities.

We derive from Fourier inequalities between weighted Lebesgue spaces, weighted Sobolev gradient inequalities for a wide range of indices. The weight functions for which these inequalities hold are easily computable, but the norm constants are not optimal.

A semidiscrete multiplier is an operator between a space of functions or distributions on a locally compact Abelian group G on the one hand, and a space of sequences on a discrete subgroup H of G on the other hand, with the property that it commutes with shifts by H . We describe the basic form of such operators and show a number of representation theorems for classical spaces like L ^p, C _0, etc. We also point out parallels to representation theorems for multipliers.

We characterize the existence of finite tight frames whose frame elements are of predetermined length. In particular, we derive a “fundamental inequality” which completely characterizes those sequences which arise as the lengths of a tight frame’s elements. Furthermore, using concepts from classical physics, we show that this characterization has an intuitive physical interpretation.

The Balian-Low Theorem is one of many manifestations of the uncertainty principle in harmonic analysis. Originally stated as a result on the poor time-frequency localization of generating functions of Gabor orthonormal bases, it has become a synonym for many general and abstract problems in time-frequency analysis. In this chapter we present some of the directions in which the Balian-Low Theorem has been extended in recent years.

We define the distributional Zak transform and study some of its properties. We show how the distributional Zak transform can be used as an effective tool in the theory of Gabor systems where the window function belongs to the Schwartz class S (ℝ) and where the product of the parameters defining the Gabor system is rational. In particular, we obtain a necessary and sufficient condition for the linear span of such a Gabor system to be dense in S (ℝ) in the topology of S (ℝ) and, if this is the case, we show that a dual window in the Schwartz class can be constructed. We also characterize when such a Gabor system satisfies the Riesz property.

Gabor duality studies have resulted in a number of characterizations of dual Gabor frames, among which the Wexler-Raz identity and the operator approach reformulation by Janssen and by Daubechies, Landau, and Landau are well known. A concise overview of existing Gabor duality characterizations is presented. In particular, we demonstrate that the Gabor duality conditions by Wexler and Raz [ 23 ] and by Daubechies, Landau, and Landau [ 6 ], and the parametric dual Gabor formula of [ 15 ] are equivalent.

Pseudodifferential operators are an indispensable tool for the study of partial differential equations and are therefore a branch of classical analysis. In this chapter we offer an approach using time-frequency methods. In this approach time-frequency representations that are standard in signal analysis are used to set up the formalism of pseudodifferential operators, and certain classes of function spaces and symbols, the modulation spaces, arise naturally in the investigation. Although the approach is “pedestrian” and based more on engineering intuition than on “hard” analysis, strong results on boundedness and Schatten class properties are within its scope.

This chapter is an introduction to an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L ^2 function. Background and motivation for the conjecture are provided in the form of a survey of related ideas, results, and open problems in frames, Gabor systems, and other aspects of time-frequency analysis, especially those related to independence. The partial results that are known to hold for the conjecture are also presented and discussed.

We consider two classes of actions on ℝ^n—one continuous and one discrete. For matrices of the form A = e ^B with B ∈ M _n(ℝ), we consider the action given by γ → γ A ^t. We characterize the matrices A for which there is a cross-section for this action. The discrete action we consider is given by γ → γ A ^k, where A ∈ GL _n(ℝ). We characterize the matrices A for which there exists a cross-section for this action as well. We also characterize those A for which there exist special types of cross-sections; namely, bounded cross-sections and finite-measure cross-sections. Explicit examples of cross-sections are provided for each of the cases in which cross-sections exist. Finally, these explicit cross-sections are used to characterize those matrices for which there exist minimally supported frequency (MSF) wavelets with infinitely many wavelet functions. Along the way, we generalize a well-known aspect of the theory of shift-invariant spaces to shift-invariant spaces with infinitely many generators.