Convolution Equations and Singular Integral Operators: Selected Papers (Operator Theory: Advances and Applications, 206)

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. Title Page Copyright Page Table of Contents Preface Introduction References Inversion of Finite Toeplitz Matrices 1. Inversion of Toeplitz matrices 2. Inverse problem 3. Continual analogue References Inversion of Finite Toeplitz Matrices Consisting of Elements of a Noncommutative Algebra 1. Theorems on inversion of Toeplitz matrices 2. Properties of solutions of equations 3. Inverse problem 4. Other theorems on inversion of Toeplitz matrices 5. Properties of solutions of equations 6. Inverse problem for equations References Matrix Integral Operators on a Finite Interval with Kernels Depending on the Difference of the Arguments 1. Two lemmas 2. The inversion formula 2.1. The main result of this section is the following. 2.2. Formula 3. Properties of the solutions of main equations 4. Inverse problem References The Resultant Matrix and its Generalizations. I. The Resultant Operator for Matrix Polynomials 1. Lemma on multiple extensions of systems of vectors 1.1. 1.2. 2. Auxiliary propositions 2.1. 2.2. 2.3. 3. Main theorem 3.1. 3.2. 3.3. 4. Applications 4.1. 4.2. 5. Kernel of Bezoutian 5.1. 5.2. References The Resultant Matrix and its Generalizations. II. The Continual Analogue of the Resultant Operator 1. Formulation of the main theorem 1.1. 1.2. 1.3. 1.4. 1.5. 2. A lemma 3. Proof of the main theorem 4. Scalar case 4.1. 4.2. 5. Applications 5.1. 5.2. 5.3. 6. Continual analogue of the Bezoutian References The Spectrum of Singular Integral Operators in Lp Spaces 1. The spectrum of discrete Wiener-Hopf equations in hp spaces 1.1. 1.2. 1.3. 2. The spectrum of singular integral operators in Lp spaces 3. Estimate for the norm of the singular integral operator 4. The spectrum of singular integral operators in symmetric spaces References On an Algebra Generated by the Toeplitz Matrices in the Spaces hp 1. Toeplitz operators in the spaces hp 1.1. 1.2. 2. Algebra generated by the Toeplitz operators 2.1. 2.2. References On Singular Integral Equations with Unbounded Coefficients 1. Auxiliary propositions 2. On the boundedness of the operator of singular integration 3. Operators of the form 4. Operators of the form References Singular Integral Equations with Continuous Coefficients on a Composed Contour 1. Auxiliary propositions and theorems on solvability 2. Quotient algebra 3. Normal solvability and index of operators in the algebra References On a Local Principle and Algebras Generated by Toeplitz Matrices 1. Localizing classes 1.1. 1.2. 1.3. 2. First example 2.1. 2.2. 3. Some properties of operators generated by Toeplitz matrices in lp spaces 3.1. 3.2. 3.3. 3.4. 4. Second example and its applications 4.1. 4.2. 5. Inversion of Toeplitz matrices 5.1. 5.2. 5.3. 6. Algebra of operators with Fredholm symbol 7. Algebras generated by paired operators 7.1. 7.2. References The Symbol of Singular Integral Operators on a Composed Contour 1. Auxiliary propositions 1.1. 1.2. 1.3. 2. Singular operators with coefficients in Λ+ and Λ− 3. Algebra generated by singular operators with coefficients in Λ+m and Λ−m 4. Singular integral operators with coefficients in Λm 5. Symbols of singular operators 5.1. 5.2. 6. Concluding remarks 6.1. 6.2. References One-dimensional Singular Integral Operators with Shift Introduction 1. Auxiliary statements 2. Main statement 3. Example References Algebras of Singular Integral Operators with Shift 1. 2. References Annotation This volume contains 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.The papers are both of theoretical and numerical interest.In particular, the papers examine fast algorithms for inversion of these operators, the theory of discrete and continuous resultants, inversion via factorization, and symbol construction. Originally the papers were written in Russian more than thirty years ago; their English translation is published here for the first time. These papers solved difficult problems and opened new venues in the above-mentioned areas. They are still frequently quoted, and moreover, they exert a continuing influence on numerical analysis and other areas of Pure and Applied Mathematics and Engineering.The book is addressed to a wide audience of mathematicians and engineers, from graduate students to researchers, whose interests lie in the above-mentioned areas The papers were originally written in Russian more than thirty years ago; their English translation is published here for the first time. These papers solved difficult problems and opened new venues in the above-mentioned areas. They are still frequently quoted, and moreover, they exert a continuing influence on numerical analysis and other areas of pure and applied mathematics and engineering. The book is addressed to a wide audience of mathematicians and engineers, from graduate students to researchers, whose interests lie in the above-mentioned areas --Book Jacket This volume contains 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. The papers are both of theoretical and numerical interest, in particular, they examine fast algorithms for inversion of these operators, the theory of discrete and continuous resultants, inversion via factorization, and symbol construction