A wavelet theory for local fields and related groups

Using a prime element of a local field K of positive characteristic p, the concepts of multiresolution analysis (MRA) and wavelet can be generalized to such a field. We prove a version of the splitting lemma for this setup and using this lemma we have constructed the wavelet packets associated with such MRAs. We show that these wavelet packets generate an orthonormal basis by translations only. We also prove an analogue of splitting lemma for frames and construct the wavelet ...

The aim of this exposition is to explain basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$. After developing a general concept for compact groups and their homogeneous spaces we give concrete examples for tori -which reflect the situation on $R^n$- and for spheres $S^2$ and $S^3...

Shah and Abdullah [Complex Analysis Operator Theory, 9 (2015), 1589-1608] have introduced a generalized notion of nonuniform multiresolution analysis (NUMRA) on local field $K$ of positive characteristic in which the translation set $\Lambda$ acting on the scaling function to generate the core space $V_{0}$ is no longer a group, but is the union of ${\mathcal Z}$ and a translate of ${\mathcal Z}$, given by $\Lambda=\left\{0,u(r)/N \right\}+{\mathcal Z}$, where $N \ge 1$ is an...

In this article, we present a constructive method for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and group theory.

In this paper a countable family of new compactly supported {\em non-Haar} $p$-adic wavelet bases in ${\cL}^2(\bQ_p^n)$ is constructed. We use the wavelet bases in the following applications: in the theory of $p$-adic pseudo-differential operators and equations. Namely, we study the connections between wavelet analysis and spectral analysis of $p$-adic pseudo-differential operators. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differ...

We consider a class of semidirect products $G = \mathbb{R}^n \rtimes H$, with $H$ a suitably chosen abelian matrix group. The choice of $H$ ensures that there is a wavelet inversion formula, and we are looking for criteria to decide under which conditions there exists a wavelet such that the associated reproducing kernel is integrable. It is well-known that the existence of integrable wavelet coefficients is related to the question whether the unitary dual of $G$ contains o...

In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let $\Gamma_j$, $j \in J$, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group $G$ and study systems of the form $\cup_{j \in J}\{g_{j,p}(\cdot - \gamma)\}_{\gamma \in \Gamma_j, p \in P_j}$ with generators $g_{j,p}$ in $L^2(G)$ and with each $P_j$ being a countable or an unco...

The general construction of frames of p-adic wavelets is described. We consider the orbit of a mean zero generic locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss relation of this result to the multiresolution wavelet analysis.

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this article we discuss the construction of frames for $L^2(\R^n)$ using the action of closed subgroups $H\subset \mathrm{GL}(n,\mathbb{R})$ such that $H$ has an open orbit $\cO$ in $\R^n$ under the action $(h,\omega)\mapsto (h^{-1})^T(\omega)$. If $H$ has the form $ANR$, where $A$ is simply connected and abelian, $N$ contains a co-compact discrete subgrou...

This paper is devoted to wavelet analysis on adele ring $\bA$ and the theory of pseudo-differential operators. We develop the technique which gives the possibility to generalize finite-dimensional results of wavelet analysis to the case of adeles $\bA$ by using infinite tensor products of Hilbert spaces. The adele ring is roughly speaking a subring of the direct product of all possible ($p$-adic and Archimedean) completions $\bQ_p$ of the field of rational numbers $\bQ$ with ...