Narrow Results

In this paper, we study Gabor frames in the matrix-valued signal space $L^2(G,{ℂ}^{n\times n})$, where $G$ is a locally compact abelian group which is metrizable and $\sigma$-compact, and $n$ is a positive integer. First, we give sufficient conditions on scalars in an infinite combination of vectors (from a given matrix-valued Gabor frame) to constitute a new frame for the space $L^2(G,{ℂ}^{n\times n})$. This generalizes a result due to Aldroubi. Second, we discuss frame conditions for finite sums of matrix-valued Gabor frames. Sufficient conditions for finite sums of matrix-valued Gabor frames in terms of frame bounds are established. It is shown that the sum of images of matrix-valued Gabor frames under bounded linear operators acting on $L^2(G,{ℂ}^{n\times n})$ constitute a frame for the space $L^2(G,{ℂ}^{n\times n})$ provided operators are adjointable with respect to the matrix-valued inner product and satisfy a majorization. Finally, we show that matrix-valued Gabor frames are stable under small perturbations.

The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Recently, Kutyniok and Strohmer generalized the notion of the Wilson system to the lattices whose generator matrix is in Hermite normal form.

We extend their result to the full characterization of Wilson orthonormal bases on the general lattice of volume 1/2. Moreover, we generalize this result to other forms of Wilson systems differing from the classical one by the appropriate sign modification.

A Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé from Gabor tight frame elements, when the redundancy of the Gabor system is 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Afterwards, Gröchenig posed a question whether the construction of an orthonormal Wilson basis is possible for a Gabor tight frame of redundancy 3. We give a partial positive answer to this question constructing in this case a Wilson system being a tight frame with bound 1.

In this paper, we focus on the construction of Wilson frames and their dual frames for general lattices of volume (K even) in the discrete-time setting. We obtain a necessary and sufficient condition for two Bessel sequences having Wilson structure to be dual frames for l²(ℤ). When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for l²(ℤ), show that a Wilson frame for l²(ℤ) can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame.

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.

For a, b > 0 and g ∈ L²(ℝ), write 𝒢(g, a, b) for the Gabor system:

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

It is well-known that the Gabor expansions converge to identity operator in weak* sense on the Wiener amalgam spaces as sampling density tends to infinity. In this paper, we prove the convergence of Gabor expansions to identity operator in the operator norm as well as weak* sense on $W(L^p,{\ell }^q)$ as the sampling density tends to infinity. Also we show the validity of the Janssen’s representation and the Wexler–Raz biorthogonality condition for Gabor frame operator on $W(L^p,{\ell }^q)$.

In digital signal and image processing one can only process discrete signals of finite length, and the space ${ℂ}^L$ is the preferred setting. Recently, Kutyniok and Strohmer constructed orthonormal Wilson bases for ${ℂ}^L$ with general lattices of volume $\frac{L}{2}$ ($L$ even). In this paper, we extend this construction to Wilson frames for ${ℂ}^L$ with general lattices of volume $\frac{L}{2K}$, where $K\in \mathbb{N}$ and $L\in 2K\mathbb{N}$. We obtain a necessary and sufficient condition for two sequences having Wilson structure to be dual frames for ${ℂ}^L$. When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for ${ℂ}^L$, show that a Wilson frame for ${ℂ}^L$ can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying 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.

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.

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.

Recently, Gabor analysis on locally compact abelian (LCA) groups has become the focus of an active research. In practice, the time variable cannot be negative. The half real line ${ℝ}_{+}=(0,\infty )$ is an LCA group under multiplication and the usual topology, with the Haar measure $d\mu =\frac{dx}{x}$. This paper addresses Gabor frame multipliers and Parseval duals for $L^2({ℝ}_{+},d\mu )$. We introduce and characterize Gabor frame multipliers and Parseval Gabor frame multipliers based on Zak transform matrices. Our Zak transform matrix is essentially different from the conventional Zibulski–Zeevi matrix. It allows us to define Gabor frame generators by designing suitable matrix-valued functions of finite size. We also prove that an arbitrary Gabor frame ${𝒢}(g,a,b)$ admits a Parseval dual frame/tight dual frame whenever $lna\cdot lnb$ are rational numbers not greater than $\frac{1}{2}$.

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.

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.