On the properties of systems of root functions of a second-order discontinuous operator
type: Item
instance of: scholarly article
title: On the properties of systems of root functions of a second-order discontinuous operator
zbMATH Open document ID: 1192.34100
DOI: 10.1134/S1064562409040176
author: N. E. Zubov
published in: Doklady Mathematics
publication date: 2010-02-05T00:00:00Z
review text: Let \(G=\cup^m_{l=1}G_l\), \(G_l=(\xi_{l-1},\xi_l)\), \(a=\xi_0<\xi_1<\dots <\xi_m=b\), where \(-\infty<a<b<+\infty\). By \(W^n_r(G)\), \(1\leq r\leq\infty\), is denoted the space of functions possessing the property that, if \(f\in W^n_r(G)\) then, for every \(G_l\), \(l=1,2,\dots,m\), there exists a function \(f_l\) from the Sobolev space \(W^n_r(G_l)\) such that \(f(x)=f_l(x)\) for \(\xi_{l-1}<x<\xi_l\). On \(G\), the formal differential operator \(Lu=u''+p(x)u'+q(x)u\) with complex-valued coefficients \(p\in L_r(G)\), \(1<r\leq 2\), and \(q\in L_1(G)\) is considered. A system of root functions of the operator \(L\) is an arbitrary system \(\{u_k\}_{k=1}^{\infty}\) of complex-valued identically nonzero functions from \(W_1^2(G)\) that satisfy a.e. in \(G\) the equation \(Lu_k+\lambda_ku_k=\theta_ku_{k-1}\), where \(\theta_k\) is either \(0\) (in this case \(u_k\) is called an eigenfunction) or \(1\) (in this case it is required that \(\lambda_k=\lambda_{k-1}\) and \(u_k\) is called an associated function of order \(j\), where \(\theta_k=\theta_{k-1}=\dots=\theta_{k-j+1}=1\) and \(\theta_{k-j}=0\)). The authors establish a necessary and sufficient condition for the Riesz basis property of a system of root functions of the operator \(L\) and then analyze the absolute and uniform convergence and the rate of uniform convergence of biorthogonal expansions of discontinuous functions in terms of root functions of the operator \(L\) (for \(p(x)\equiv 0\)). Finally, some concrete examples of boundary value operators are presented.
reviewed by: Yakov Yakubov
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 5666526
zbMATH Keywords: discontinuous operator
zbMATH Keywords: root functions
zbMATH Keywords: Sobolev space
zbMATH Keywords: Riesz basis
zbMATH Keywords: biorthogonal expansion
MaRDI profile type: Publication
cites work: Uniform convergence of biorthogonal series for the Schrödinger operator with multipoint boundary conditions
cites work: Equiconvergence of biorthogonal expansions in root functions of differential operators. II
cites work: A generalized Bessel inequality for ordinary differential operators with nonsmooth coefficients and a generalization of the Riesz theorem
full work available at URL: s1064562409040176
OpenAlex ID: W1970401014
description: scientific article; zbMATH DE number 5666526