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Sergey Lukomskii

Publications and source records attributed to Sergey Lukomskii.

6 recordsLinked to original sources
arXiv2023

On approximation by tight wavelet frames on Vilenkin groups

We consider the approximate properties of tight wavelet frames on Vilenkin group $G$. Let $\{G_n\}_{n\in \mathbb{Z} }$ be a main chain of subgroups, $X$ be a set of characters. We define a step function $λ(χ)$ that is constant on cosets ${G}_n^\bot\setminus{G}_{n-1}^\bot$ by equalities $λ({G}_n^\bot\setminus{G}_{n-1}^\bot)=λ_n>0$ for which $\sum\frac{1}{λ_n}<\infty$. We find the order of approximation of functions $f$ for which $\int_X|λ( χ)\hat{f}(χ)|^2dν(χ)<\infty$. As a corollary, we obtain an approximation error for functions from Sobolev spaces with logarithmic weight.

math.FA↗
arXiv2022

p-adic tight wavelet frames

We propose a simple method to construct step mask and corresponding step wavelet functions that generate tight wavelet frames on the field of p-adic numbers. To construct tight wavelet frames we do not use the principle of unitary extension, we use Pontryagin's principle of duality.

math.FA↗
arXiv2020

Numerical solution of linear differential equations with discontinuous coefficients and Henstock integral

In this article we consider the problem of approximative solution of linear differential equations $y'+p(x)y=q(x)$ with discontinuous coefficients $p$ and $q$. We assume that coefficients of such equation are Henstock integrable functions. To find the approximative solution we change the original Cauchy problem to another problem with piecewise-constant coefficients. The sharp solution of this new problems is the approximative solution of the original Cauchy problem. We find the degree approximation in terms of modulus of continuity $ω_δ(P),\ ω_δ(Q)$, where $P$ and $Q$ are $f$-primitive for coefficients $p$ and $q$.

math.CA↗
arXiv2014

Non-Haar MRA on local fields of positive characteristic

We propose a simple method to construct integral periodic mask and corresponding scaling step functions that generate non-Haar orthogonal MRA on the local field $ F^{(s)}$ of positive characteristic $p$. To construct this mask we use two new ideas. First, we consider local field as vector space over the finite field $GF(p^s)$. Second, we construct scaling function by arbitrary tree that has $p^s$ vertices. By fixed prime number $p$ there exist $p^{s(p^s-2)}$ such trees.

math.NT↗