Narrow Results

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 $L^2(\mathrm{\Omega })$. 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.

In this paper, we introduce the concept of weak Gabor bi-frame (WGBF) in a general closed subspace ${ℳ}$ of $L^2(ℝ)$. It is a generalization of Gabor bi-frame, and is new even if ${ℳ}=L^2(ℝ)$. A WGBF for ${ℳ}$ contains all information of ${ℳ}$ to some extent. Let $a$, $b>0$, and $S$ be an $a\mathbb{Z}$-periodic subset of $ℝ$ with positive measure. This paper is devoted to characterizing WGBFs for $L^2(S)$ of the form

In this paper, we study multivariate Gabor frames in matrix-valued signal spaces over locally compact abelian (LCA) groups, where the lower frame condition depends on a bounded linear operator $\mathrm{\Theta }$ on the underlying matrix-valued signal space. This type of Gabor frame is also known as a multivariate $\mathrm{\Theta }$-Gabor frame. By extending work of G$\check{\mathrm{\text{a}}}$vruta, we present necessary and sufficient conditions for the existence of $\mathrm{\Theta }$-Gabor frames of multivariate matrix-valued Gabor systems. Some operators which can transform multivariate matrix-valued Gabor and $\mathrm{\Theta }$-Gabor frames into $\mathrm{\Theta }$-Gabor frames in terms of adjointable operators are discussed. Finally, we give a Paley–Wiener-type perturbation result for multivariate matrix-valued $\mathrm{\Theta }$-Gabor frames.

This paper addresses vector-valued subspace Gabor frames with rational time-frequency product. By introduction of a suitable Zak transform matrix, we characterize vector-valued subspace Gabor frames, Riesz bases and orthonormal bases. We characterize the uniqueness of Gabor duals of type I and type II. Using the uniqueness results, we extend the classical Balian-Low theorem to vector-valued subspace Gabor frames. We also point out Ron-Shen duality principe does not hold for a general vector-valued subspace Gabor frame.

The theory of vector-valued frames (also called superframes) has important applications in signal multiplexing, and has interested many mathematicians in recent years. In this paper, we introduce the notion of weak Gabor duals of type II under the setting of subspaces $L^2(S,{ℂ}^L)$, where $S$ is a periodic subset of $ℝ$. Using the Zak transform matrix method, we characterize weak Gabor duals of type II and their uniqueness. An example is also presented to illustrate the generality of our results.

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.

A dual Gabor window pair has two functions that can reconstruct any function in $L^2(ℝ)$ using certain systems of their modulated and translated forms. Few explicit examples of dual Gabor window pairs are known. This paper constructs explicit examples with trigonometric form in one dimension as well as higher dimensions. Also, in the discrete time domain, the trigonometric form allows us to evaluate the Gabor coefficients efficiently using the Discrete Fourier Transform. The windows have fixed support and arbitrary smoothness.