We find necessary and sufficient conditions on refinable step function under which this function generates an orthogonal MRA in the $L_{2}(\mathfrak{G})$
 -spaces on Vilenkin group $\mathfrak{G}$
 . We consider a class of refinable step functions for which the mask m
 0(χ) is constant on cosets $\mathfrak{G}_{-1}^{\bot}\chi$
 and its modulus |m
 0(χ)| has two values only: 0 and 1. We prove that any refinable step function φ from this class that generates an orthogonal MRA on Vilenkin group $\mathfrak{G}$
 has Fourier transform with condition $\operatorname{supp}\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$
 . We show the sharpness of this result, too.

Step Refinable Functions and Orthogonal MRA on Vilenkin Groups

A generalization of Mallat’s classic theory of multiresolution analysis (MRA) on local fields of positive characteristic was considered by Jiang et al. (J Math Anal Appl 294:523–532, 2004). In this paper, we present a notion of nonuniform MRA on local field $K$ of positive characteristic. The associated subspace $V_0$ of $L^2(K)$ has an orthonormal basis, a collection of translates of the scaling function $\varphi $ of the form $\{ \varphi (x-\lambda ) \}_{ \lambda \in \Lambda }$ where $\Lambda = \{ 0,r/N \}+ \mathcal{Z}, \,N \ge 1$ is an integer and $r$ is an odd integer such that $r$ and $N$ are relatively prime and $\mathcal{Z}=\{u(n): n\in \mathbb {N}_{0}\}$. We obtain the necessary and sufficient condition for the existence of associated wavelets and present an algorithm for the construction of nonuniform MRA on local fields starting from a low-pass filter $m_{0}$ with appropriate conditions.

Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic

We provide a characterization of wavelets on local fields of positive characteristic based on results on affine and quasi-affine frames. This result generalizes the characterization of wavelets on Euclidean spaces by means of two basic equations. We also give another characterization of wavelets. Further, all wavelets which are associated with a multiresolution analysis on such a local field are also characterized.

Characterization of wavelets and MRA wavelets on local fields of positive characteristic

An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.

Wavelet frames on Vilenkin groups and their approximation properties

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.

http://vm-rggru.narod.ru/Teach/YA/Farkov_Rodionov_2011.pdf

Algorithms for wavelet construction on Vilenkin groups

Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

/pdf/nonstationary-wavelets-related-to-the-walsh-functions-4jo7pmv08f.pdf

Evgeny A. Rodionov

Papers

Algorithms for wavelet construction on Vilenkin groups

Nonstationary Wavelets Related to the Walsh Functions