Constructions of MRA-based wavelets and frames in Walsh analysis (Q2825724)

The author reviews MRA-based constructions of orthonormal wavelet bases and tight frames defined on Vilenkin groups. In Section 2, Lang's wavelets on \(\mathbb{R}_+\) are described. In Section 3, orthogonal wavelets on Vilenkin groups are discussed in detail. The author recalls how to design orthogonal wavelets using the framework of multiresolution analysis (MRA). The result is summarized in Algorithm A. The set of all compactly supported step refinable functions is characterized in Section 3 as well. One more approach for the design of scaling functions (Theorem 3.10) is interpreted in the context of Algorithm A in Remark 3.11. In Section 4, a method to design an MRA-based Parseval wavelet frame is discussed. The result is presented in Algorithm B. In the article, the difference between wavelets defined on Vilenkin groups and on the real line is indicated. The difference concerns the smoothness and the order of approximation. The theoretical parts of the article are accompanied by various examples of wavelet and scaling functions. The paper contains a fairly complete reference list concerning the topic.