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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.

We consider a class of (N, M)-elementary step functions on the p-adic Vilenkin group. We prove that (N, M)-elementary step function generates a MRA on p-adic Vilenkin group if and only if it is generated by a special N-valid rooted tree on the set of vertices {0,1,…p - 1} with the vector (0,…,0) ∈ ℤ^N as a root.

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products $U{P}_{\lambda }$ and $U{P}_G$ that are similar to the Heisenberg uncertainty product. $U{P}_{\lambda }$ and $U{P}_G$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $U{P}_{\lambda }$ and $U{P}_G$ in terms of Walsh and Haar basis are given.

Wavelet sets provide a range of wavelets that can be analyzed with ease. The constructions of wavelet sets and scaling sets in different settings are studied in several papers in the literature. Wavelet sets provide a good source of examples and counterexamples in wavelet theory. In this paper we have studied properties of (multi)wavelet sets and associated wavelets for Vilenkin group. Further, results related to scaling sets and generalized scaling sets are given along with some characterizations.