Related papers: Riesz Wavelets, Tiling and Spectral Sets in LCA Gr…

We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gr\"ochenig, Kutyniok, and Seip in the article:…

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G=Q_p, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H=Z_p, the ring of…

W. C. Lang determined wavelets on Cantor dyadic group by using Multiresolution analysis method. In this paper we have given characterization of wavelet sets on Cantor dyadic group providing another method for the construction of wavelets.…

Single wavelet sets, and thus single wavelets, are shown to exist for the actions of all crystallographic groups on $\mathbb R^2$ under all integer dilations. Examples of such sets satisfying the additional requirement that they are finite…

Using the wavelet theory introduced by the author and J. Benedetto, we present examples of wavelets on p-adic fields and other locally compact abelian groups with compact open subgroups. We observe that in this setting, the Haar and Shannon…

Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that every subset of a finite abelian group has…

By using ergodic theoretic techniques following Hillel F\"{u}rstenberg, we prove that measurable subsets of a locally compact abelian group of positive upper density contain Szemer\'{e}di-wise configurations defined by an arbitrary compact…

We establish a general form of Wiener's lemma for measures on locally compact abelian (LCA) groups by using Fourier analysis and the theory of F{{\o}}lner sequences. Our approach provides a unified framework that that encompasses both the…

Following work of Rieffel, we define the Cayley compactification of an abelian group with specified generating set. We investigate its structure using methods from discrete geometry and commutative algebra.

We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to…

Motivated by the recent work of Bownik and Ross \cite{BR}, and Jakobsen and Lemvig \cite{JL}, this article generalizes latest results on reproducing formulas for generalized translation invariant (GTI) systems to the setting of super-spaces…

We characterize all the locally compact abelian (LCA) groups that contain quasicrystals (a class of model sets). Moreover, we describe all possible quasicrystals in the group constructing an appropriate lattice associated with the cut and…

A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called locally precompact. Within the class of locally…

In this paper, we analyse the circumstances in which the adjoint Gabor system is an R-dual of a given Gabor frame in the context of separable uniform time-frequency lattices in locally compact abelian groups. In this regard, we also prove a…

In this paper a sampling theory for unitary invariant subspaces associated to locally compact abelian (LCA) groups is deduced. Working in the LCA group context allows to obtain, in a unified way, sampling results valid for a wide range of…

A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface…

This paper establishes two fundamental results on the existence of exponential Riesz basis in non-Archimedean locally compact Abelian groups: the existence of Riesz basis of exponentials for all finite unions of balls and the non-existence…

In this paper we study the category LCA(2) of certain non-locally compact abelian topological groups, and extend the notion of Weil index. As applications we deduce some product formulas for curves over local fields and arithmetic surfaces.

We obtain a set of generalized eigenvectors that provides a generalized spectral decomposition for a given unitary representation of a commutative, locally compact topological group. These generalized eigenvectors are functionals belonging…

A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A…