Functional analysis and geometry. Selim Grigorievich Krein centennial

Cover......Page 1
Title page......Page 4
Contents......Page 10
Preface......Page 12
 1. Selim Grigorievich Krein and his mathematics......Page 14
 3. List of 83 S. Krein’s PhD students......Page 18
 References......Page 19
My father Selim Krein......Page 30
Kyiv, Fall of 1943 through 1946. The rebirth of mathematics......Page 36
Selim Gregorievich Krein in Stony Brook......Page 42
 References......Page 44
 1. Introduction......Page 46
 2. Preliminaries......Page 49
 3. Proof of Theorem 1.5......Page 51
 References......Page 57
 1. Introduction......Page 58
 2. Preliminaries......Page 60
 3. Some characterizations of the class of r.i. spaces satisfying Dunford-Pettis criterion of weak compactness......Page 62
 4. Orlicz spaces satisfying Dunford-Pettis criterion of weak compactness......Page 64
 5. The martingale difference Cesaro mean property in r.i. spaces......Page 68
 References......Page 70
 1. Introduction......Page 74
 3. Examples......Page 77
 4. Applications......Page 78
 5. More examples and open questions......Page 83
 References......Page 84
 1. Introduction......Page 86
 2. Definition of a hypercomplex system with a locally compact basis......Page 87
 3. The results......Page 90
 References......Page 93
 1. Introduction......Page 94
 2. Preliminaries......Page 95
 3. Horospherical radon transforms......Page 97
 4. The totally geodesic Radon transforms......Page 102
 Acknowledgment......Page 106
 References......Page 107
Total positivity, Grassmannian and modified Bessel functions......Page 110
 1. Introduction......Page 111
 2. Positivity. Proof of Theorem 1.6......Page 115
 References......Page 119
A remark on the intersection of plane curves......Page 122
 Introduction......Page 123
 1. Focal loci......Page 125
 2. Double planes......Page 126
 3. The main result......Page 130
 4. Genera of subvarieties: a survey......Page 134
 References......Page 137
 2. Outline of the general theory......Page 142
 3. Topological integrable billiards......Page 146
 4. Fomenko’s conjecture. Modeling of any nondegenerate integrable system of general form with two degrees of freedom by an integrable topological billiard......Page 147
 5. The part �� of Fomenko’s conjecture is correct. Kharcheva-Vedyushkina’s theorem......Page 149
 6. The topology of the Liouville foliation of the billiards with nonconvex angles.......Page 152
 7. Integrable billiards in the Minkowski space......Page 155
 References......Page 159
 1. Introduction......Page 162
 2. Formulation of the results......Page 164
 3. Cayley–Hamilton Theorem revisited......Page 168
 4. Countably generated \QQ-vector spaces......Page 173
 5. Bosonic vertex operators......Page 175
 Acknowledgments......Page 177
 References......Page 178
 1. Dual spherical polynomials (following Maxwell)......Page 180
 2. The Cauchy- Radon transform in ℝⁿ......Page 181
 3. The horospherical Cauchy transform on the sphere......Page 183
 5. Relation to Maxwell polynomials......Page 185
 References......Page 186
 1. Introduction......Page 188
 2. Boundedness of ��_{��}......Page 191
 3. The operator ��²_{��}......Page 196
 References......Page 197
 Introduction......Page 198
 1. Danielewski’s Surfaces......Page 199
 2. Gizatullin surfaces......Page 200
 3. Danilov-Gizatullin surfaces......Page 201
 4. \G_{��}-actions on normal surfaces......Page 202
 6. Amalgams......Page 204
 7. Generalized Gizatullin surfaces......Page 205
 References......Page 208
 Introduction......Page 210
 1. Formulation of main results......Page 211
 2. Differential identities for some elementary functions......Page 213
 3. Proofs......Page 215
 4. Final remarks and conclusions......Page 218
 References......Page 219
 1. Introduction......Page 222
 2. Notations and our use of the language of sheaves......Page 223
 3. Topological criteria for global holomorphic similarity......Page 224
 4. Bumps on Riemann surfaces......Page 227
 5. ��-adapted pairs of compact sets on 1-dimensional complex spaces......Page 232
 6. Jordan stable points......Page 234
 7. Proof of Theorem 1.2......Page 237
 References......Page 243
 1. Introduction......Page 246
 2. Weighted geometric means......Page 250
 3. A self-dual geometric mean......Page 256
 4. Additional remarks......Page 260
 References......Page 261
 1. Introduction and main results......Page 264
 2. Lie groups and their representations......Page 267
 3. Hardy-Steklov operator associated with operators ��₁,��₂,...,��_{��}.......Page 270
 4. Interpolation spaces......Page 274
 5. Approximation by Hardy-Steklov averages, K-functor and modulus of continuity......Page 275
 6. Shannon sampling, Paley-Wiener frames and abstract Besov subspaces......Page 277
 7. Besov subspaces in Hilbert spaces......Page 280
 8. Applications......Page 282
 References......Page 283
 1. Introduction......Page 286
  2.1. Polyanalytic and true polyanalytic spaces......Page 288
  2.2. The structure of the Fock spaces.......Page 289
  2.3. The structure of the Bergman spaces on the half-plane and on the disk.......Page 290
 3. Toeplitz operators with distributional symbols......Page 291
  3.1. Toeplitz operators in Bergman spaces on the disk and on the half-plane with symbols being derivatives of ��–C measures.......Page 292
  3.2. Toeplitz operators in \Fc with symbols being coderivatives of ��–FC measures.......Page 294
  4.1. Transformation of Toeplitz operators in the true poly-Bergman space on the half-plane.......Page 295
  4.2. Transformation of Toeplitz operators in the true poly-Bergman space on the disk.......Page 296
  4.3. Transformation of Toeplitz operators in the true poly-Fock space.......Page 298
   5.1.2. Uniqueness......Page 299
  5.2. Boundedness......Page 300
 References......Page 302
 1. Introduction......Page 304
 2. Space ��ₐₑ......Page 305
 3. Differentiation......Page 308
 4. Compositions......Page 311
 References......Page 312
Back Cover......Page 314