Vilenkin group

Necessary and Sufficient Condition for an Orthogonal Scaling Function on Vilenkin Groups

Authors: Berdnikov G. S.

There are several approaches to the problem of construction of an orthogonal MRA on Vilenkin groups, but all of them are reduced to the search of the so-called scaling function. In 2005 Yu. Farkov used the so-called “blocked sets” in order to find all possible band-limited scaling functions with compact support for each set of certain parameters and his conditions are necessary and sufficient. S. F. Lukomskii, Iu. S. Kruss and G. S.

Graphs with Contours in Multiresolution Analysis on Vilenkin Groups

Authors: Berdnikov G. S.

The aim of this article is to study the problem of constructing mutiresolution analysis on Vilenkin group. Previous papers by S. F. Lukomskii, Iu. S. Kruss and the author present an algorithm for constructing scaling functions ϕ with compact support, Fourier transform of which also has compact support. The description of such algorithm is tightly connected with directed graphs of special structure, which are constructed with the help of so-called N-valid trees. One of the special properties of these graphs is the absence of directed cycles — contours.

On the Existence of Continual Closed U-set

Authors: Yurchenko I. S.

In this work we consider a system of characters of the Vilrnkin group G and study uniqueness sets for series for system of character of Vilenkin group (in other words, U-sets). We prove a sufficient condition for the U-set on the Vilenkin group and constructed continual closed U-set on the Vilenkin group.