Graphs with contours in multiresolution analysis on Vilenkin groups

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 $\varphi$ 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. This property allowsadmits the construction of scaling functions $\varphi$, Fourier transform of which has compact support. This approach has a number of advantages. Firstly, this algorithm does not include exhaustive search in contrast to algorithm using the noton of “blocked sets”, which is described in the papers by Yu. A. Farkov. Secondly, such approach is conveniently generalized to the case of local fields of positive characteristic, which was done in the papers by Kruss Iu. S. The contents of the current paper represents the first step of using digraphs with contours for similar purpose. Taking further the ideas of previous research we construct digraph with only one simple contour using $1$-valid tree. It appears that such graph also generates a scaling function $\varphi$. However, since the contour appears, such scaling function's Fourier transform does not have compact support.