A wavelet theory for local fields and related groups

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H < {\rm GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classificatio...

We present an algorithm for construction step wavelets on local fields of positive characteristic.

Let $G$ be a Vilenkin group. In 2008, Y. A. Farkov constructed wavelets on $G$ via the multiresolution analysis method. In this article, a characterization of wavelet sets on $G$ is established, which provides another method for the construction of wavelets. As an application, the relation between multiresolution analyses and wavelets determined from wavelet sets is also presented. To some extent, these results positively answer a question mentioned by P. Mahapatra and D. Sin...

We consider the coorbit theory associated to general continuous wavelet transforms arising from a square-integrable, irreducible quasi-regular representation of a semidirect product group $G = \mathbb{R}^d \rtimes H$. The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assump...

This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.

The purpose of this informal article is to introduce the reader to some of the objects and methods of the theory of p-adic representations. My hope is that students and mathematicians who are new to the subject will find it useful as a starting point. It consists mostly of an expanded version of the notes for my two lectures at the "Dwork trimester" in June 2001.

We described a wide class of $p$-adic refinable equations generating $p$-adic multiresolution analysis. A method for the construction of $p$-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example, which gives a new 3-adic wavelet basis. Another realization leads to the $p$-adic Haar bases which were known before.

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say $N$, and we show that the wavelet filters are indexed by the infinite dimensional group $G$ of functions from $X$ into the...

We describe how the Deaconu-Renault groupoid may be used in the study of wavelets and fractals.

This work introduces author's approach to harmonic analysis on algebraic groups over functional two-dimensional local fields. For a two-dimensional local field a Hecke algebra which is formed by operators which integrate pro-locally-constant complex functions over a non-compact domain is defined and its properties are described.