Blaschke products and their applications

Cover......Page 1
Blaschke Products and Their Applications......Page 3
Preface......Page 6
Contents......Page 8
1 Introduction......Page 10
2 Spectra of Toeplitz Operators......Page 12
3 Compositions with Blaschke Products and the Ap Condition......Page 17
4 More on the Spectra of Toeplitz Operators......Page 21
5 Modelling of Monotone Functions with the Help of Blaschke Products......Page 25
6 Applications to the KdV Equation......Page 36
7 Some Open Problems......Page 37
References......Page 38
1 Introduction......Page 40
2 Symmetric Function Theory......Page 42
3 Frequent Hypercyclicity......Page 44
4 Proof of Theorem 3......Page 47
5 Proof of Theorem 4......Page 48
References......Page 50
A Survey on Blaschke-Oscillatory Differential Equations, with Updates......Page 52
2 Introduction......Page 53
3.1 Measurements for the Quantity of Points......Page 57
3.2 Measurements for the Density of Points......Page 58
4.1 Improvement of (4)......Page 60
4.2 Basic Properties of Solutions......Page 63
4.3 Solutions in F\N are Possible......Page 66
5.1 Pointwise Growth Restrictions for A(z)......Page 68
5.2 Integrated Growth Restrictions for A(z)......Page 71
6 Univalent Coefficient Function......Page 74
7.1 Pointwise Estimates......Page 80
7.2 Integrated Estimates......Page 82
8 One Infinite Prescribed Zero Sequence......Page 85
9 Two Infinite Prescribed Zero Sequences......Page 87
9.1 New Result Involving Two Sequences......Page 88
9.2 Interpolation in the BMOA Space......Page 89
9.3 Wiman-Valiron Theory in D......Page 90
9.4 Proof of Theorem 21......Page 91
10 Finite Prescribed Zero Sequences......Page 93
11 Blaschke-Oscillatory Equations of Arbitrary Order......Page 94
12.1 Zeros Versus Critical Points......Page 98
12.2 Necessary and Sufficient Conditions......Page 99
12.3 Prescribed Critical Points......Page 101
13 Concluding Remarks......Page 102
References......Page 104
1 Introduction......Page 108
2 Two Systems of Analytic Functions in H2+......Page 109
3 Bi-orthogonal Expansions in H2+......Page 112
4 Main Results and Their Proofs......Page 116
References......Page 121
1 Introduction......Page 122
2 A Two-Sided Blaschke Sequence......Page 123
Open Question......Page 124
3 A Surjective Composition Operator on KB......Page 125
References......Page 127
1 Introduction......Page 128
2 Definitions and Known Results......Page 129
3 The Primitive of a CT Function......Page 132
4 The Connection to Univalent Functions......Page 133
5 Connection to BMOA and Teichmuller Space......Page 134
References......Page 140
1 Critical Points of Bounded Analytic Functions......Page 141
2.1 Conformal Metrics and Developing Maps......Page 143
2.2 Maximal Conformal Pseudometrics and Maximal Blaschke Products......Page 146
3 Some Properties of Maximal Blaschke Products......Page 149
3.1 Schwarz' Lemmas......Page 150
3.2 Related Extremal Problems in Hardy and Bergman Spaces......Page 152
3.3 Boundary Behaviour of Maximal Blaschke Products......Page 154
4 The Gauss Curvature PDE and the Berger-Nirenberg Problem......Page 157
Berger-Nirenberg Problem......Page 158
References......Page 163
1 Introduction......Page 166
2.1 Growth of Nevanlinna Characteristic and Zero Distribution......Page 167
2.2 Factorization of Classes Defined by the Growth of T(r,f)......Page 169
3.1 Growth of the Maximum Modulus and Zero Distribution......Page 170
3.2 Factorization of Classes Defined by the Growth of logM(r,f)......Page 171
4 A Concept of rhoinfty-Order......Page 173
5 Proof of Theorem 10......Page 175
References......Page 179
1 Introduction......Page 181
2 Derivations of Formulae for Delta2(r, Bn)......Page 182
3 Comparison of Hardy and Sampling Means......Page 185
4 Location of Zeroes......Page 188
References......Page 192
1 Preliminaries......Page 193
2 The Main Theorem......Page 196
References......Page 198
Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane......Page 199
1.1 Affine Multiresolution Analysis......Page 200
1.2 The Blaschke Group......Page 201
1.3 The Hardy Space of the Upper Half Plane......Page 202
2.1 A Special Lattice in the Upper Half Plane......Page 204
2.2 Multiresolution on the Upper Half Plane......Page 205
2.3 The Projection Operator Corresponding to the nth Resolution Level......Page 209
2.4 Reconstruction Algorithm......Page 210
References......Page 212
1 Introduction......Page 215
2.1 Finite Blaschke Products......Page 216
2.4 The Bloch Space......Page 217
2.5 Analytic Besov Spaces......Page 218
2.7 Composition Operators on Conformally Invariant Spaces......Page 219
3 Norms of Composition Operators on Quotient Besov Spaces......Page 220
4.1 Norms on the True Bloch Space......Page 223
4.2 Norms on the True Dirichlet Space......Page 225
References......Page 227
1 Introduction......Page 229
2.1 Computable Functions and Sets......Page 230
2.2 Computable vs. Computably Enumerable Sets......Page 232
2.3 From Whence All This Comes: The Fundamental Theorem of Computability Theory......Page 233
2.4 Uniform vs. Non-uniform Computability......Page 234
3 Computable Analysis on the Unit Disk......Page 235
4 Some Basic Computability Results on Blaschke Products......Page 238
5 The Missing Parameter......Page 245
6 Interpolating Sequences and Naftalévich's Theorem......Page 248
7 Inner Functions-Frostman's Theorem......Page 249
8 Inner Functions-Factorization......Page 251
References......Page 253
1 Introduction......Page 255
2 Elementary Results......Page 256
2.1.2 Critical Points and Critical Values......Page 257
2.1.3 Factorizations of Polynomials......Page 258
2.2.1 Some Simple Properties......Page 259
2.2.2 Critical Points and Critical Values......Page 260
2.2.3 Factorizations of Finite Blaschke Products......Page 261
3.1.1 Definitions and Some Basic Properties......Page 262
3.1.2 Monodromy......Page 263
3.1.4 The Julia Set......Page 264
3.2.1 Definition and Some Basic Properties......Page 265
Jacobi Elliptic Functions......Page 266
Definition and Properties of Chebyshev-Blaschke Products......Page 267
Chebyshev-Blaschke Products of Small Degrees......Page 268
3.2.2 Monodromy......Page 269
3.2.4 The Julia Set......Page 270
3.2.5 The Approximation Problems......Page 271
New Approximation Problems......Page 272
4.1.2 Polynomials That Share a Set......Page 273
4.1.3 Dinh's Result......Page 274
4.2.1 Finite Blaschke Products That Share Two Values in the Unit Disk......Page 275
4.2.2 Finite Blaschke Products That Share a Set......Page 276
References......Page 277
1 Introduction......Page 280
2.1 Basic Notation......Page 282
2.2 Model Spaces......Page 283
2.4 Kernel Functions, Conjugation, and Angular Derivatives......Page 284
2.5 Two Results of Aleksandrov......Page 286
2.6 The Compressed Shift......Page 287
2.7 Clark Unitary Operators and Their Spectral Measures......Page 288
2.8 Finite Dimensional Model Spaces......Page 290
2.9 Truncated Toeplitz Operators......Page 291
3 Tu as a Linear Space......Page 294
4 Norms of Truncated Toeplitz Operators......Page 296
5 The Bounded Symbol and Related Problems......Page 300
6 The Spatial Isomorphism Problem......Page 305
7 Algebras of Truncated Toeplitz Operators......Page 306
8 Truncated Toeplitz C*-Algebras......Page 309
9 Unitary Equivalence to a Truncated Toeplitz Operator......Page 311
10 Unbounded Truncated Toeplitz Operators......Page 314
11 Smoothing Properties of Truncated Toeplitz Operators......Page 317
12 Nearly Invariant Subspaces......Page 318
References......Page 320