Convolution Equations and Singular Integral Operators (eBook)

Title Page 4
Copyright Page 5
Table of Contents 6
Preface 7
Introduction 8
References 17
Inversion of Finite Toeplitz Matrices 22
1. Inversion of Toeplitz matrices 22
2. Inverse problem 24
3. Continual analogue 26
References 27
Inversion of Finite Toeplitz Matrices Consisting of Elements of a Noncommutative Algebra 28
1. Theorems on inversion of Toeplitz matrices 28
2. Properties of solutions of equations 36
3. Inverse problem 44
4. Other theorems on inversion of Toeplitz matrices 49
5. Properties of solutions of equations 58
6. Inverse problem for equations 62
References 66
Matrix Integral Operators on a Finite Interval with Kernels Depending on the Difference of the Arguments 68
1. Two lemmas 69
2. The inversion formula 72
2.1. The main result of this section is the following. 72
2.2. Formula 76
3. Properties of the solutions of main equations 77
4. Inverse problem 81
References 84
The Resultant Matrix and its Generalizations. I. The Resultant Operator for Matrix Polynomials 85
1. Lemma on multiple extensions of systems of vectors 86
1.1. 86
1.2. 87
2. Auxiliary propositions 93
2.1. 93
2.2. 95
2.3. 97
3. Main theorem 98
3.1. 98
3.2. 99
3.3. 101
4. Applications 102
4.1. 102
4.2. 104
5. Kernel of Bezoutian 105
5.1. 105
5.2. 107
References 108
The Resultant Matrix and its Generalizations. II. The Continual Analogue of the Resultant Operator 109
1. Formulation of the main theorem 110
1.1. 110
1.2. 111
1.3. 113
1.4. 113
1.5. 114
2. A lemma 115
3. Proof of the main theorem 118
4. Scalar case 121
4.1. 121
4.2. 123
5. Applications 123
5.1. 123
5.2. 126
5.3. 126
6. Continual analogue of the Bezoutian 126
References 128
The Spectrum of Singular Integral Operators in Lp Spaces 130
1. The spectrum of discrete Wiener-Hopf equations in hp spaces 131
1.1. 131
1.2. 131
1.3. 132
2. The spectrum of singular integral operators in Lp spaces 133
3. Estimate for the norm of the singular integral operator 139
4. The spectrum of singular integral operators in symmetric spaces 141
References 143
On an Algebra Generated by the Toeplitz Matrices in the Spaces hp 145
1. Toeplitz operators in the spaces hp 146
1.1. 146
1.2. 147
2. Algebra generated by the Toeplitz operators 148
2.1. 148
2.2. 150
References 151
On Singular Integral Equations with Unbounded Coefficients 152
1. Auxiliary propositions 153
2. On the boundedness of the operator of singular integration 155
3. Operators of the form 157
4. Operators of the form 159
References 161
Singular Integral Equations with Continuous Coefficients on a Composed Contour 162
1. Auxiliary propositions and theorems on solvability 164
2. Quotient algebra 169
3. Normal solvability and index of operators in the algebra 171
References 173
On a Local Principle and Algebras Generated by Toeplitz Matrices 174
1. Localizing classes 175
1.1. 175
1.2. 175
1.3. 176
2. First example 177
2.1. 177
2.2. 178
3. Some properties of operators generated by Toeplitz matrices in lp spaces 179
3.1. 179
3.2. 180
3.3. 181
3.4. 182
4. Second example and its applications 183
4.1. 183
4.2. 184
5. Inversion of Toeplitz matrices 186
5.1. 186
5.2. 188
5.3. 191
6. Algebra of operators with Fredholm symbol 192
7. Algebras generated by paired operators 195
7.1. 195
7.2. 198
References 199
The Symbol of Singular Integral Operators on a Composed Contour 202
1. Auxiliary propositions 203
1.1. 203
1.2. 204
1.3. 205
2. Singular operators with coefficients in .+ and .- 205
3. Algebra generated by singular operators with coefficients in .+m and .-m 207
4. Singular integral operators with coefficients in .m 209
5. Symbols of singular operators 211
5.1. 211
5.2. 213
6. Concluding remarks 214
6.1. 214
6.2. 215
References 215
One-dimensional Singular Integral Operators with Shift 217
Introduction 217
1. Auxiliary statements 219
2. Main statement 222
3. Example 225
References 226
Algebras of Singular Integral Operators with Shift 228
1. 228
2. 231
References 232