Partial Gabor frames and dual frames

The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.

On the Parity Under Metapletic Operators and an Extension of a Result of Lyubarskii and Nes

In this work we show that if the frame property of a Gabor frame with window in Feichtinger’s algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger’s algebra and any separable lattice of density $$\tfrac{n+1}{n}$$n+1n, $$n \in \mathbb {N}_+$$n∈N+, cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.

Multigenerator Gabor Frames on Local Fields

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.

Gabor frames on non-Archimedean fields

In this paper, we introduce the concept of periodic Gabor frames on non-Archimedean fields of positive characteristic. We first establish a necessary and sufficient condition for a periodic Gabor system to be a Gabor frame for [Formula: see text]. Then, we present some equivalent characterizations of Parseval Gabor frames on non-Archimedean fields by means of some fundamental equations in the time domain. Finally, potential applications of Gabor frames on non-Archimedean fields are also discussed.

Sampling time-frequency localized functions and constructing localized time-frequency frames

We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.

ON HILBERT TRANSFORM OF GABOR AND WILSON SYSTEMS

Given that the Gabor system {EmbTnag}m,n∈ℤ is a Gabor frame for L2(ℝ), a sufficient condition is obtained for the Gabor system {EmbTnaHg}m,n∈ℤ to be a Gabor frame, where Hg denotes the Hilbert transform of g ∈ L2(ℝ). It is proved that the Hilbert transform operator and the frame operator for the Gabor Bessel sequence {EmbTnag}m,n∈ℤ commute with each other under certain conditions. Also, a sufficient condition is obtained for the Wilson system [Formula: see text] to be a Wilson frame given that [Formula: see text] is a Wilson frame. Finally, we obtain conditions under which the Hilbert transform operator and the frame operator for the Wilson Bessel sequence [Formula: see text] commute with each other.

RATIONAL TIME-FREQUENCY GABOR FRAMES ASSOCIATED WITH PERIODIC SUBSETS OF THE REAL LINE

For a, b > 0 and g ∈ L2(ℝ), write 𝒢(g, a, b) for the Gabor system: [Formula: see text] Let S be an aℤ-periodic measurable subset of ℝ with positive measure. It is well-known that the projection 𝒢(gχS, a, b) of a frame 𝒢(g, a, b) in L2(ℝ) onto L2(S) is a frame for L2(S). However, when ab > 1 and S ≠ ℝ, 𝒢(g, a, b) cannot be a frame in L2(ℝ) for any g ∈ L2(ℝ), while it is possible that there exists some g such that 𝒢(g, a, b) is a frame for L2(S). So the projections of Gabor frames in L2(ℝ) onto L2(S) cannot cover all Gabor frames in L2(S). This paper considers Gabor systems in L2(S). In order to use the Zak transform, we only consider the case where the product ab is a rational number. With the help of a suitable Zak transform matrix, we characterize Gabor frames for L2(S) of the form 𝒢(g, a, b), and obtain an expression for the canonical dual of a Gabor frame. We also characterize the uniqueness of Gabor duals of type I (respectively, type II).

The Stability and Perturbation of Bivariate Gabor Frames and Applications in Material Engineering

Material science broadly encompasses the fundamental study of solid matter with the goal of engineering new materials with superior properties, and ultimately enabling altogether new types of devices The window functions and bivariate Gabor frames are introduced. The existence of bivariate Gabor frames with compact support is discussed. Sufficient conditions for irregular bivariate Gabor system to be frames are presented by means of frame multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal bivariate vector-valued wavelets with compact support is proposed, and their properties are investigated. The pyramid decomposition scheme is derived based on a generalized multiresolution structure.

New Proof for Balian-Low Theorem of Nonlinear Gabor System

The main purpose of this paper is to give a new proof of the Balian-Low theorem for Gabor system{eimθ(2πt)g(t−n), m,n∈ℤ}, which is proposed by Fu et al. and associated with nonlinear Fourier atoms. To this end, we establish the relationships between spacesL2(ℝ,dθ)andL2(ℝ). We also introduce the concept of frame associated with nonlinear Fourier atoms forL2(ℝ,dθ)and obtain many subsidiary results for this kind of (Gabor) frames.