Relations between convolution type operators on intervals and on the half-line
type: Item
instance of: scholarly article
title: Relations between convolution type operators on intervals and on the half-line
zbMATH Open document ID: 0974.47023
DOI: 10.1007/BF01192422
published in: Integral Equations and Operator Theory
publication date: 2001-11-30T00:00:00Z
review text: The main object of the paper is an operator of the form \({\mathcal W}_{\Phi_{\mathcal A}, \Omega}= r_{\Omega} {\mathcal A}_{|\widetilde{H}^{r,p}(\Omega)} : \widetilde{H}^{r,p}(\Omega) \rightarrow H^{s,p}(\Omega)\), acting between Bessel potential spaces, where \(r,s \in {\mathbb R}\), \(p \in (1,\infty)\), \(\Omega\) is a finite union of intervals on \(\mathbb R\), \({\mathcal A}= {\mathcal F}^{-1} \Phi_{\mathcal A} {\mathcal F}^{-1}\) is a convolution type operator from \(H^{r,p}(\mathbb R)\) into \(H^{s,p}(\mathbb R)\), and \(r_{\Omega}\) is the restriction of distributions from \({\mathcal S}'({\mathbb R})\) to \(\Omega\). The operator \({\mathcal W}_{\Phi_{\mathcal A}, \Omega}\) and a matrix Wiener-Hopf operator \(W_{\Phi}\) on the half axis \({\mathbb R}_+\) are said to be matricially coupled if the following matrix equality holds \[ \begin{pmatrix} {\mathcal W}_{\Phi_{\mathcal A}, \Omega} & 0 \\ 0 & I_Y\end{pmatrix} = E \begin{pmatrix} W_{\Phi} & 0 \\ 0 & I_Z\end{pmatrix} F, \] where \(Y\), \(Z\) are Banach spaces, and \(E\), \(F\) are invertible bounded linear operators. One refers to the above matrix equality as to the equivalence after extension relation. The main result of the paper is a method to construct equivalence after extension relations for arbitrary unions of intervals and orders \(k= r,s \in {\mathbb R}\) which are not critical, i.e., \(k-1/p \notin {\mathbb Z}\). Some applications are given as well.
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A20
zbMATH DE Number: 1507906
zbMATH Keywords: Wiener-Hopf operator
zbMATH Keywords: Bessel potential space
zbMATH Keywords: equivalence after extension relation
zbMATH Keywords: Fredholm properties
zbMATH Keywords: convolution type operator
zbMATH Keywords: matricially coupled operators
author: L. P. Castro
author: Frank-Olme Speck
MaRDI profile type: Publication
cites work: Q3331715
cites work: Q4287397
cites work: Convolution equations of the first kind on a finite interval in Sobolev spaces
cites work: Wiener-Hopf operators with oscillating symbols and convolution operators on a union of intervals
cites work: Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility
cites work: Finite Interval Convolution Operators on the Bessel Potential Spaces H
cites work: The corona theorem and the canonical factorization of triangular AP matrix functions -- Effective criteria and explicit formulas
cites work: Meromorphic Factorization, Partial Index Estimates and Elastodynamic Diffraction Problems1)
cites work: On the Characterization of the Intermediate Space in Generalized Factorizations
cites work: On the inversion of higher order Wiener-Hopf operators
cites work: A fredholm study for convolution operators with piecewise continuous symbols on a union of a finite and a semi-infinite intervel1
cites work: Regularity properties and generalized inverses of delta-related operators
cites work: The diffraction problem for a half plane with different face impedances revisited
cites work: The Sommerfeld problem revisited: Solution spaces and the edge conditions
cites work: Q3866607
cites work: Q3907885
cites work: Q3310334
cites work: Q3027610
cites work: FACTORIZATION OF ALMOST PERIODIC MATRIX-VALUED FUNCTIONS AND THE NOETHER THEORY FOR CERTAIN CLASSES OF EQUATIONS OF CONVOLUTION TYPE
cites work: Q4317777
cites work: Q4721089
cites work: A note on first kind convolution equations on a finite interval
cites work: On convolution equations with semi-almost periodic symbols on a finite interval
cites work: On the diffraction problem of a half plane with different face impedances.
cites work: Q3031337
cites work: Q4332726
cites work: Q3837859
cites work: Minimal normalization of Wiener-Hopf operators in spaces of Bessel potentials
cites work: Q4040220
cites work: Convolution equations on a finite segment and factorization of elliptic matrices
cites work: Q3915444
cites work: A GENERALIZATION OF THE WIENER-HOPF METHOD FOR CONVOLUTION EQUATIONS ON A FINITE INTERVAL WITH SYMBOLS HAVING POWER-LIKE ASYMPTOTICS AT INFINITY
cites work: Q3821119
cites work: Q4282408
cites work: Q4893982
cites work: Sommerfeld Half-plane Problems with Higher Order Boundary Conditions
cites work: Q5735487
cites work: Q3715553
cites work: Q5522602
cites work: Q4332722
cites work: Q4003552
cites work: Solution of Nonlinear Equations by Iterative Procedures which Use Approximation Techniques
full work available at URL: https://doi.org/10.1007/bf01192422
OpenAlex ID: W1984760483
description: scientific article; zbMATH DE number 1507906