Riesz Wavelets, Tiling and Spectral Sets in LCA Groups

In this paper we prove that wavelet expansions on the Cantor dyadic group G converge unconditianally in the dyadic Hardy space H1 (G). We will do it for wavelets satisfying the regularity condition of Hölder-Lipshitz type.

Advances in Mathematics: Scientific Journal

We construct a wavelet frame system on locally compact abelian (LCA) group G associated with the multiresolution analysis and Haar measures. We show the characterization of the wavelet frame set and the scaling sequence on L 2 (G). The dilation and translation of wavelet frame sets for time-frequency localization in LCA groups have been set up. We obtain an orthonormal wavelet basis for L 2 (G) using the scaling sequence. We also establish the relationship between multiresolution analysis and wavelet functions. Finally, we obtain periodization for the multiresolution analysis using time-frequency localization on a periodic wavelet frame. The periodization holds wavelets' regular properties and decay conditions.

Contemporary Mathematics, 2006

Most of the examples of wavelet sets are for dilation sets which are groups. We find a necessary and sufficient condition under which subspace wavelet sets exist for dilation sets of the form AB, which are not necessarily groups. We explain the construction by a few examples. 1991 Mathematics Subject Classification. 42C40,43A85.

2012

The multiresolution analysis (MRA) on certain non-abelian locally compact groups G is considered. Characterizations for a refinable function to generate an MRA in L2(G) are given. Here, no regularity properties or decay conditions are placed on the scaling functions. MRAs for L2(G) generated by a self-similar tile as a scaling function are shown and Haar-like wavelet bases are constructed. Concrete examples related to Heisenberg group are provided to illustrate the theorems.

Journal of the Australian Mathematical Society, 1969

P-Adic Numbers, Ultrametric Analysis, and Applications, 2011

In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.

Journal of Functional Analysis, 1997

A spectral set is a subset of R n with Lebesgue measure 0 < () < 1 such that there exists a set of exponential functions which form an orthogonal basis of L 2 (). The spectral set conjecture of B. Fuglede states that a set is a spectral set if and only if tiles R n by translation. We study sets which tile R n using a rational periodic tile set S = Z n + A, where A 1 N 1 Z 1 Nn Z is nite. We characterize geometrically sets that tile R n with such a tile set. Certain tile sets S have the property that every bounded measurable set which tiles R n with S is a spectral set, with a xed spectrum S. We call such S a universal spectrum for such S. We give a necessary and su cient condition for a rational periodic set to be a universal spectrum for S, which is expressed in terms of factorizations A B = G where G = Z N 1 Z Nn , and A := A (mod Z n). In dimension n = 1 we show that S has a universal spectrum whenever N 1 is the order of a \good" group in the sense of Haj os, and for various other sets S.

Journal of Fourier Analysis and Applications, 2006

Let G be the semidirect product group of a separable locally compact unimodular group N of type I with a closed subgroup H of Aut(N). The group N is not necessarily commutative. We consider irreducible subrepresentations of the unitary representation of G realized naturally on L 2 (N), and investigate the wavelet transforms associated to them. Furthermore, the irreducible subspaces are characterized by certain singular integrals on N analogous to the Cauchy-Szegö integral.

Contemporary mathematics, 2008

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 wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there is a natural notion of a dilation-reflection wavelet set. We prove that dilationreflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilation-translation wavelet theory. There are certain measurable sets which can serve simultaneously as dilation-translation wavelet sets and dilation-reflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different.

Mathematical Notes, 2007

We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with p n numerical coefficients generate multiresolution analyses in L 2 (G). It is noted that the coefficients of these scaling equations can be calculated from the given values of p n parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L 2 (G) is stable and has a linearly independent system of "integer" shifts. We present several examples illustrating these results.