The equivalence of causal and ordinary invertibility for integral convolution operators
type: Item
instance of: scholarly article
title: The equivalence of causal and ordinary invertibility for integral convolution operators
zbMATH Open document ID: 1022.47032
DOI: 10.1023/A:1012538216183
author: V. A. Skopin
published in: Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: Let \(S\) be a cone in \(\mathbb{R}^n\). A linear operator \(T\) acting in a function space on \(\mathbb{R}^n\) is said to be causal if the value \((Tx)(t)\) at an arbitrary point \(t \in \mathbb{R}^n\) is completely determined by the values of \(x\) on the set \(t-S\). An operator \(T\) is said to be causally invertible if the inverse \(T^{-1}\) exists and is also causal. It is shown that for the integral convolution operator \[ (Gx)(t)=\int_S g(s) x(t-s) ds \] with \(g \in L^1(S)\), the ordinary invertibility of the operator \(I+G\) in \(L^p(S)\) coincides with the causal invertibility for \(n\geq 2\).
reviewed by: Boris Rubin
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 1915938
zbMATH Keywords: convolution operators
zbMATH Keywords: causal invertibility
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1915938