Convolution Equations and Singular Integral Operators

This book offers English translations of 13 groundbreaking papers on Toeplitz matrices and Wiener-Hopf equations and other classes of discrete and continuous convolution operators and singular integral equations. Both are of theoretical and numerical interest.

This book consists of translations into English of several pioneering papers in the areas of discrete and continuous convolution operators and on the theory of singular integral operators published originally in Russian. The papers were wr- ten more than thirty years ago, but time showed their importance and growing in?uence in pure and applied mathematics and engineering. The book is divided into two parts. The ?rst ?ve papers, written by I. Gohberg and G. Heinig, form the ?rst part. They are related to the inversion of ?nite block Toeplitz matrices and their continuous analogs (direct and inverse problems) and the theory of discrete and continuous resultants. The second part consists of eight papers by I. Gohberg and N. Krupnik. They are devoted to the theory of one dimensional singular integral operators with discontinuous co- cients on various spaces. Special attention is paid to localization theory, structure of the symbol, and equations with shifts. ThisbookgivesanEnglishspeakingreaderauniqueopportunitytogetfam- iarized with groundbreaking work on the theory of Toepliz matrices and singular integral operators which by now have become classical. In the process of the preparation of the book the translator and the editors took care of several misprints and unessential misstatements. The editors would like to thank the translator A. Karlovich for the thorough job he has done. Our work on this book was started when Israel Gohberg was still alive. We see this book as our tribute to a great mathematician.

The papers selected for this volume have never been translated into English Closes a gap in the English literature on the subject The material can be used for advanced courses and seminars in mathematics and engineering Includes supplementary material: sn.pub/extras

Inhalt
Inversion of Finite Toeplitz Matrices.- Inversion of Finite ToeplitzMatrices Consisting of Elements of a Noncommutative Algebra.- Matrix Integral Operators on a Finite Interval with Kernels Depending on the Difference of the Arguments.- The Resultant Matrix and its Generalizations. I. The Resultant Operator for Matrix Polynomials.- The Resultant Matrix and its Generalizations. II. The Continual Analogue of the Resultant Operator.- The Spectrum of Singular Integral Operators in L p Spaces.- On an Algebra Generated by the Toeplitz Matrices in the Spaces h p .- On Singular Integral Equations with Unbounded Coefficients.- Singular Integral Equations with Continuous Coefficients on a Composed Contour.- On a Local Principle and Algebras Generated by Toeplitz Matrices.- The Symbol of Singular Integral Operators on a Composed Contour.- One-dimensional Singular Integral Operators with Shift.- Algebras of Singular Integral Operators with Shift.