arXiv subjects

S. F. Lukomskii

Publications and source records attributed to S. F. Lukomskii.

6 recordsLinked to original sources

arXiv2023

On approximation by tight wavelet frames on the field of $p$-adic numbers

We discuss the problem on approximation by tight step wavelet frames on the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $G_n=\{x=\sum_{k=n}^\infty x_k p^k\}$, $X$ 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 $\int_X|λ( χ)\hat{f}(χ)|^2dν(χ)<\infty$

math.NT↗

arXiv2017

Discrete Riesz MRA on local fields of positive characteristic

We propose a method to construct Riesz MRA on local fields of positive characteristic and corresponding scaling step functions that generate it.

math.FA↗

arXiv2017

Riesz Bounds of Spline Affine Systems

We construct a family of spline affine Riesz bases, i.e. sequences of dilations and translations generated by the special spline functions $ψ_m$, and we prove that their Riesz bounds are independent of $m$. We put $ψ_0=χ$ as the Haar step function and every following function $ψ_{m+1}$ is obtained by integrating the previous function $ψ_m$ with consequent antiperiodization. We give a representation of the spline $ψ_m$ as a finite sum of Rademacher chaos series and we use a notion of a simple Walsh spectrum of a function in connection with orthogonality of affine systems.

math.FA↗

arXiv2014

N-valid trees in wavelet theory on Vilenkin groups

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 iff it is generated by a special $N$-valid rooted tree on the set of vertices $\{0,1,\dots p-1\}$ with the vector $(0,...,0)\in \mathbb Z^N$ as a root. Bibliography: 15 titles.

math.FA↗

arXiv2013

Trees in Wavelet analysis on Vilenkin groups

We consider a class of $(1,M)$-elementary step functions on the $p$-adic Vilenkin group. We prove that $(1,M)$-elementary step function generates a MRA on $p$-adic Vilenkin group iff it is generated by a rooted tree on the set of vertices $\{0,1,\dots p-1\}$ with 0 as a root. Bibliography: 14 titles.

math.FA↗

arXiv2012

Step refinable functions and orthogonal MRA on $p$-adic Vilenkin groups

We find the necessary and sufficient conditions for refinable step function under which this function generates an orthogonal MRA in the $L_2(\mathfrak G)$ -spaces on Vilenkin groups $\mathfrak G$. We consider a class of refinable step functions for which the mask $m_0(χ)$ is constant on cosets $\mathfrak G_{-1}^\bot$ and its modulus $|m_0(χ)|$ takes two values only: 0 and 1. We will prove that any refinable step function $φ$ from this class that generates an orthogonal MRA on $p$-adic Vilenkin group $\mathfrak G$ has Fourier transform with condition ${\rm supp} \hat φ(χ)\subset \mathfrak G_{p-2}^\bot$. We show the sharpness of this result too.

math.FA↗