Convolution Equations and Singular Integral Operators

1. Front Matter

   Pages i-vii

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2. Introduction

   • Leonid Lerer, Vadim Olshevsky, Ilya Spitkovsky

   Pages ix-xxii

3. Inversion of Finite Toeplitz Matrices

   • Israel Gohberg, Georg Heinig

   Pages 1-6

4. Inversion of Finite ToeplitzMatrices Consisting of Elements of a Noncommutative Algebra

   • Israel Gohberg, Georg Heinig

   Pages 7-46

5. Matrix Integral Operators on a Finite Interval with Kernels Depending on the Difference of the Arguments

   • Israel Gohberg, Georg Heinig

   Pages 47-63

6. The Resultant Matrix and its Generalizations. I. The Resultant Operator for Matrix Polynomials

   • Israel Gohberg, Georg Heinig

   Pages 65-88

7. The Resultant Matrix and its Generalizations. II. The Continual Analogue of the Resultant Operator

   • Israel Gohberg, Georg Heinig

   Pages 89-109

8. The Spectrum of Singular Integral Operators in L _p Spaces

   • Israel Gohberg, Nahum Krupnik

   Pages 111-125

9. On an Algebra Generated by the Toeplitz Matrices in the Spaces h _p

   • Israel Gohberg, Nahum Krupnik

   Pages 127-133

10. On Singular Integral Equations with Unbounded Coefficients

    • Israel Gohberg, Nahum Krupnik

    Pages 135-144

11. Singular Integral Equations with Continuous Coefficients on a Composed Contour

    • Israel Gohberg, Nahum Krupnik

    Pages 145-156

12. On a Local Principle and Algebras Generated by Toeplitz Matrices

    • Israel Gohberg, Nahum Krupnik

    Pages 157-184

13. The Symbol of Singular Integral Operators on a Composed Contour

    • Israel Gohberg, Nahum Krupnik

    Pages 185-199

14. One-dimensional Singular Integral Operators with Shift

    • Israel Gohberg, Nahum Krupnik

    Pages 201-211

15. Algebras of Singular Integral Operators with Shift

    • Israel Gohberg, Nahum Krupnik

    Pages 213-217

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.

• Factor
• Finite
• Integral equation
• Singular integral
• Volume
• algorithms
• applied mathematics
• commutative property
• construction
• convolution
• kernel
• mathematics
• operator
• singular integral equation
• time

• Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Israel

  Leonid Lerer

• Department of Mathematics, University of Connecticut, Storrs, USA

  Vadim Olshevsky

• Department of Mathematics, College of William & Mary, Williamsburg, USA

  Ilya M. Spitkovsky

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