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 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2016, Volume 21, Issue 3, Pages 39–56 (Mi fpm1733)

Complete systems of eigenfunctions of the Vladimirov operator in $L^{2}(B_r)$ and $L^{2}(\mathbb{Q}_{p})$

A. Kh. Bikulova, A. P. Zubarevbc

a N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow
b Samara State Aerospace University
c Samara State Transport University

Abstract: We construct new bases of real functions from $L^{2}(B_{r})$ and from $L^{2}(\mathbb{Q}_{p})$. These functions are eigenfunctions of the $p$-adic pseudo-differential Vladimirov operator, which is defined on a compact set $B_{r}\subset\mathbb{Q}_{p}$ of the field of $p$-adic numbers $\mathbb{Q}_{p}$ or, respectively, on the entire field $\mathbb{Q}_{p}$. A relation between the basis of functions from $L^{2}(\mathbb{Q}_{p})$ and the basis of $p$-adic wavelets from $L^{2}(\mathbb{Q}_{p})$ is found. As an application, we consider the solution of the Cauchy problem with the initial condition on a compact set for a pseudo-differential equation with a general pseudo-differential operator that is diagonal in the basis constructed.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation

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Document Type: Article
UDC: 512.625+517.518.34+517.983.37+517.984.57

Citation: A. Kh. Bikulov, A. P. Zubarev, “Complete systems of eigenfunctions of the Vladimirov operator in $L^{2}(B_r)$ and $L^{2}(\mathbb{Q}_{p})$”, Fundam. Prikl. Mat., 21:3 (2016), 39–56

Citation in format AMSBIB
\Bibitem{BikZub16} \by A.~Kh.~Bikulov, A.~P.~Zubarev \paper Complete systems of eigenfunctions of the Vladimirov operator in $L^{2}(B_r)$ and $L^{2}(\mathbb{Q}_{p})$ \jour Fundam. Prikl. Mat. \yr 2016 \vol 21 \issue 3 \pages 39--56 \mathnet{http://mi.mathnet.ru/fpm1733}