Fractional Cauchy transform

Rita A. Hibschweiler, Thomas H. MacGregor1584885602, 9781584885603

Presenting new results along with research spanning five decades, Fractional Cauchy Transforms provides a full treatment of the topic, from its roots in classical complex analysis to its current state. Self-contained, it includes introductory material and classical results, such as those associated with complex-valued measures on the unit circle, that form the basis of the developments that follow. The authors focus on concrete analytic questions, with functional analysis providing the general framework. After examining basic properties, the authors study integral means and relationships between the fractional Cauchy transforms and the Hardy and Dirichlet spaces. They then study radial and nontangential limits, followed by chapters devoted to multipliers, composition operators, and univalent functions. The final chapter gives an analytic characterization of the family of Cauchy transforms when considered as functions defined in the complement of the unit circle. About the authors: Rita A. Hibschweiler is a Professor in the Department of Mathematics and Statistics at the University of New Hampshire, Durham, USA. Thomas H. MacGregor is Professor Emeritus, State University of New York at Albany and a Research Associate at Bowdoin College, Brunswick, Maine, USA.

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Table of contents :
c5602_fm.pdf……Page 1
FRACTIONAL CAUCHY TRANSFORMS……Page 3
Table of Contents……Page 5
List of Symbols……Page 9
Preface……Page 12
Acknowledgements……Page 15
Dedication……Page 16
References……Page 17
Preamble……Page 26
Table of Contents……Page 0
Theorem 1.1……Page 29
Theorem 1.2……Page 33
Theorem 1.4……Page 35
Corollary 1.5……Page 36
Lemma 1.7……Page 37
Theorem 1.8……Page 38
Proposition 1.9……Page 40
NOTES……Page 41
Lemma 2.1……Page 42
Lemma 2.2……Page 43
Lemma 2.4……Page 44
Lemma 2.5……Page 46
Theorem 2.7……Page 47
Theorem 2.8……Page 49
Lemma 2.9……Page 50
Theorem 2.11……Page 51
Theorem 2.12……Page 55
Lemma 2.13……Page 57
Theorem 2.14……Page 59
Lemma 2.15……Page 61
Theorem 2.16……Page 62
Theorem 2.18……Page 64
Theorem 2.21……Page 69
NOTES……Page 72
Preamble……Page 73
Theorem 3.1……Page 74
Theorem 3.3……Page 75
Proposition 3.6……Page 78
Theorem 3.8……Page 81
Corollary 3.9……Page 82
Theorem 3.10……Page 83
Theorem 3.11……Page 84
Theorem 3.12……Page 85
Lemma 3.14……Page 86
Lemma 3.18……Page 87
Lemma 3.19……Page 88
Lemma 3.20……Page 89
Theorem 3.22……Page 90
NOTES……Page 91
Preamble……Page 92
Lemma 4.1……Page 93
Theorem 4.2……Page 94
Theorem 4.3……Page 95
Lemma 4.5……Page 98
Theorem 4.6……Page 100
Proposition 4.7……Page 104
Lemma 4.8……Page 106
Theorem 4.9……Page 109
Theorem 4.10……Page 110
Theorem 4.12……Page 111
NOTES……Page 115
Preamble……Page 116
Theorem 5.1……Page 118
Theorem 5.2……Page 120
Corollary 5.3……Page 122
Theorem 5.4……Page 123
Theorem 5.5……Page 124
Theorem 5.6……Page 129
NOTES……Page 131
Preamble……Page 132
Lemma 6.1……Page 133
Theorem 6.3……Page 134
Theorem 6.4……Page 135
Theorem 6.5……Page 136
Theorem 6.6……Page 137
Theorem 6.7……Page 138
Theorem 6.8……Page 139
Theorem 6.10……Page 140
Lemma 6.11……Page 142
Theorem 6.13……Page 146
Lemma 6.14……Page 148
Lemma 6.15……Page 150
Theorem 6.16……Page 152
Corollary 6.17……Page 154
NOTES……Page 155
Preamble……Page 156
Lemma 7.1……Page 161
Theorem 7.3……Page 165
Corollary 7.6……Page 170
Theorem 7.7……Page 171
Lemma 7.9……Page 172
Theorem 7.10……Page 173
Lemma 7.11……Page 175
Theorem 7.12……Page 178
Lemma 7.14……Page 182
Theorem 7.15……Page 187
Theorem 7.17……Page 188
Theorem 7.19……Page 189
Lemma 7.21……Page 190
Lemma 7.22……Page 191
Lemma 7.23……Page 192
Theorem 7.25……Page 197
Theorem 7.30……Page 208
NOTES……Page 210
Preamble……Page 212
Theorem 8.1……Page 213
Theorem 8.3……Page 214
Corollary 8.5……Page 216
Theorem 8.7……Page 217
Lemma 8.10……Page 220
Theorem 8.11……Page 221
Theorem 8.12……Page 222
Lemma 8.13……Page 223
Theorem 8.14……Page 225
Lemma 8.15……Page 227
Theorem 8.17……Page 228
Theorem 8.18……Page 229
Corollary 8.19……Page 230
NOTES……Page 231
Preamble…….Page 232
Theorem 9.1……Page 233
Theorem 9.3……Page 236
Theorem 9.5……Page 239
Theorem 9.6……Page 241
Theorem 9.7……Page 242
Theorem 9.8……Page 244
Theorem 9.10……Page 245
NOTES……Page 248
Theorem 10.1……Page 249
Lemma 10.4……Page 254
Lemma 10.5……Page 256
Lemma 10.6……Page 258
Lemma 10.7……Page 260
Theorem 10.8……Page 264
NOTES……Page 267