Weighted Paley-Wiener theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces
type: Item
instance of: scholarly article
title: Weighted Paley-Wiener theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces
zbMATH Open document ID: 1243.65164
author: Da Xu
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-03-16T00:00:00Z
review text: The author studies the weighted \(l^{1}\) Paley-Wiener Theorem, with applications to stability of the linear multi-step time discretization for a linear Volterra-type equation of scalar type in a Hilbert space: \[ \frac{\partial u}{\partial t}+\int\limits_{0}^{t}\beta(t-s)Au(s)ds=0, \quad t>0,\quad u(0)=u_0. \] Here \(A\) is a self-adjoint positive definite linear operator in a real Hilbert space \(\mathbb{H}\) with dense domain. The positive nonincreasing kernel \(\beta\) is assumed to be the real-valued and satisfies \(\beta\in L^1(0,1)\cap C(0,\infty)\). The results extend to weighted \(l^{1}\) stability some recent \(l^{1}\) stability theorem due to the author [Numer. Math. 109, No. 4, 571--595 (2008; Zbl 1166.65401)]. Theoretical results are then applied to investigate the asymptotic behavior as \(t_n \to \infty\) of the numerical solutions of certain integro-differential equations in Hilbert space.
reviewed by: Alexander N. Tynda
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45N05
zbMATH DE Number: 6015263
zbMATH Keywords: Paley-Wiener Theorem
zbMATH Keywords: stability
zbMATH Keywords: linear multi-step time discretization
zbMATH Keywords: linear Volterra-type equation
zbMATH Keywords: integro-differential equations
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.12.035
OpenAlex ID: W2026722381
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DOI: 10.1016/J.JMAA.2011.12.035
description: scientific article; zbMATH DE number 6015263