One-Dimensional Linear Singular Integral Equations: I. Introduction

ISBN-13: 9783764325848 / Angielski / Twarda / 1992 / 269 str.

Israel Gohberg; N. Krupnik; I. Gohberg

Opis

This book is an introduction to the theory of linear one-dimensional singular integral equations. It is essentually a graduate textbook. Singular integral equations have attracted more and more attention, because, on one hand, this class of equations appears in many applications and, on the other, it is one of a few classes of equations which can be solved in explicit form. In this book material of the monograph [2] of the authors on one-dimensional singular integral operators is widely used. This monograph appeared in 1973 in Russian and later in German translation [3]. In the final text version the authors included many addenda and changes which have in essence changed character, structure and contents of the book and have, in our opinion, made it more suitable for a wider range of readers. Only the case of singular integral operators with continuous coefficients on a closed contour is considered herein. The case of discontinuous coefficients and more general contours will be considered in the second volume. We are grateful to the editor Professor G. Heinig of the volume and to the translators Dr. B. Luderer and Dr. S. Roch, and to G. Lillack, who did the typing of the manuscript, for the work they have done on this volume.

Spis treści

1 The operator of singular integration.- 1.1 Notations, definitions and auxiliary statements.- 1.1.1 The operator of singular integration.- 1.1.2 The space Lp(?,?).- 1.1.3 Interpolation theorems.- 1.2 The boundedness of the operator S? in the space Lp(?) with ? being a simple curve.- 1.3 Nonsimple curves.- 1.4 Integral operators in weighted Lp spaces.- 1.5 Unbounded curves.- 1.6 The operator of singular integration in spaces of Hölder continuous functions.- 1.7 The operator S?*.- 1.8 Exercises.- Comments and references.- 2 One-sided invertible operators.- 2.1 Direct sum of subspaces.- 2.2 The direct complement.- 2.3 Linear operators. Notations and simplest classes.- 2.4 Projectors connected with the operator of singular integration.- 2.5 One-sided invertible operators.- 2.6 Singular integral operators and related operators.- 2.7 Examples of one-sided invertible singular integral operators.- 2.8 Two lemmas on the spectrum of an element in a subalgebra of a Banach algebra.- 2.9 Subalgebras of a Banach algebra generated by one element.- 2.10 Exercises.- Comments and references.- 3 Singular integral operators with continuous coefficients.- 3.1 The index of a continuous function.- 3.2 Singular integral operators with rational coefficients.- 3.3 Factorization of functions.- 3.4 The canonical factorization in a commutative Banach algebra.- 3.5 Proof of the factorization theorem.- 3.6 The local factorization principle.- 3.7 Operators with continuous coefficients.- 3.8 Approximate solutions of singular integral equations.- 3.9 Generalized factorizations of continuous functions.- 3.10 Operators with continuous coefficients (continuation).- 3.11 Additional facts and generalizations.- 3.12 Operators with degenerating coefficients.- 3.13 A generalization of singular integral operators with continuous coefficients.- 3.14 Solution of Wiener-Hopf equations.- 3.15 Some applications.- 3.16 Exercises.- Comments and references.- 4 Fredholm operators.- 4.1 Normally solvable operators.- 4.2 The restriction of normally solvable operators.- 4.3 Perturbation of normally solvable operators.- 4.4 The normal solvability of the adjoint operator.- 4.5 Generalized invertible operators.- 4.6 Fredholm operators.- 4.7 Regularization of operators. Applications to singular integral operators.- 4.8 Index and trace.- 4.9 Functions of Fredholm operators and their index.- 4.10 The structure of the set of Fredholm operators.- 4.11 The Dependence of kerX and imX on the operator X.- 4.12 The continuity of the function kx.- 4.13 The case of a Hilbert space.- 4.14 The normal solvability of multiplication by a matrix function.- 4.15 ?±-operators.- 4.16 One-sided regularization of operators.- 4.17 Projections of invertible operators.- 4.18 Exercises.- Comments and references.- 5 Local Principles and their first applications.- 5.1 Localizing classes.- 5.2 Multipliers on $$ \mathop l\limits^ \sim _p $$.- 5.3 paired equations with continuous coefficients on $$ \mathop l\limits^ \sim _p $$.- 5.4 Operators of local type.- 5.5 Exercises.- Comments and references.- References.