From Fourier Analysis and Number Theory to Radon Transforms and Geometry: In Memory of Leon Ehrenpreis

Cover......Page 1
From Fourier Analysis and Number Theory to Radon Transforms \rand Geometry......Page 3
Preface......Page 6
Contents......Page 7
A Biography of Leon Ehrenpreis......Page 10
1 Introduction......Page 26
2 Anti-telescoping......Page 29
3 The Injection......Page 31
4 Proof of Theorem 1......Page 34
5 Proof of Theorem 2......Page 36
6 Proof of Theorem 3......Page 37
7 Proof of Theorem 4......Page 42
References......Page 44
1 Introduction......Page 46
2 First Properties of ΛE.......Page 49
3 The ``No Small Functions\'\' Condition......Page 51
4 The Linear Extremal Function for the Positivity Set of a Real Homogeneous Polynomial......Page 55
5 The Linear Extremal Function for Nonhomogeneous Real Varieties......Page 59
References......Page 64
1 Introduction......Page 65
3 Unimodular Cones......Page 68
4 Differences of Two Cones......Page 70
5 Differences of Multiple Cones......Page 71
6 Cayley Compositions......Page 72
7.1 Cones Over Hypersimplices......Page 76
References......Page 77
1 Introduction......Page 79
2 Definitions......Page 80
3 Cohomology Associated to S2k(Γ; χ, ψ) and M2k(Γ; χ, ψ)......Page 81
4 Bases of S2k(Γ; χ, ψ) and M2k(Γ; χ, ψ) for k>2......Page 84
References......Page 89
1 Introduction......Page 91
2 A Finite Version of Vinogradov\'s Inequality......Page 93
3 Commensurators and Correlators......Page 98
4 Ratner Rigidity and Moebius Disjointness......Page 102
5 Some Further Comments......Page 105
References......Page 106
1 Introduction and Statement of Results......Page 108
1.1 Differential Operators via Kloosterman Sum Symmetries......Page 111
1.2 Applications of Flipping......Page 112
1.3 Choosing a Lift......Page 115
2.1 Basic Notations and Definitions......Page 116
2.2 Poincaré Series......Page 117
2.3 Raising and Lowering Operators......Page 119
2.4 Derivatives of Poincaré Series......Page 120
2.5 Bol\'s Identity......Page 121
3 Proof of Theorems 1.1–1.3 and Corollary 1.4......Page 122
4 A Canonical Lift......Page 124
References......Page 127
Function Theory Related to the Group PSL2(R)\r......Page 130
1 Introduction......Page 131
2.1 Six Models of the Principal Series Representation......Page 136
2.2 Hyperfunctions......Page 141
2.3 The Intertwining Map V-ωs V-ω1-s......Page 144
3.1 The Space Es and Some of Its Elements......Page 146
3.2 The Green\'s Form and a Cauchy Formula for Es......Page 149
3.3 The Poisson Transformation......Page 152
3.4 Second-Order Eigenfunctions......Page 155
4.1 The Hybrid Models and the Canonical Model......Page 157
4.2 Poisson Inversion and the Canonical Model......Page 163
4.3 Differential Equations for the Canonical Model and the Sheaf of Mixed Eigenfunctions......Page 169
5.1 Spaces of Eigenfunction Germs......Page 176
5.2 The Transverse Poisson Map......Page 181
5.4 Transverse Poisson Map in the Differentiable Case......Page 189
6 Boundary Behavior of Mixed Eigenfunctions......Page 194
6.1 Interpolation Between Sheaves on H and Its Boundary......Page 195
6.2 Power Series Expansion......Page 197
6.3 Mixed Eigenfunctions Near the Diagonal of P1RxP1R\r......Page 201
6.4 The Extended Sheaf of Mixed Eigenfunctions......Page 203
6.5 Boundary Germs for the Sheaf Ds......Page 204
7.1 Analytic Case......Page 206
7.2 Differentiable Case......Page 209
A.1.1 Equivariant Eigenfunctions for the Unipotent Group N......Page 212
A.1.2 Equivariant Eigenfunctions for the Compact Group K......Page 213
A.1.3 Equivariant Eigenfunctions for the Torus A......Page 215
A.2 Poisson Transforms......Page 216
A.4 Potentials for Green\'s Forms......Page 220
A.5 Action of the Lie Algebra......Page 221
References......Page 223
1 Introduction......Page 225
2 Generalized Kimura Diffusion Operators......Page 229
3 Model Problems......Page 230
4 Perturbation Theory......Page 234
5 Main Results......Page 235
References......Page 238
1 Introduction......Page 239
2  Properties of the Matrix A......Page 241
3 Sums of lth Powers......Page 246
4 Sums of Two Primes......Page 247
5 Return of the Mobius Function......Page 248
6 Concluding Remarks......Page 253
References......Page 254
1 Introduction......Page 255
2 The Glaeser–Michael Method......Page 259
3 Computation of the Solutions......Page 264
3.1 Computation of the Glaeser Refinement......Page 268
3.2 Computation of Iterated Glaeser Refinements......Page 270
3.3 Whitney Cubes......Page 272
3.4 The Glaeser–Stable Case......Page 275
3.5 Computing the Section of a Bundle......Page 282
3.6 Computing a Continuous Solution of Linear Equations......Page 283
4 Algebraic Geometry Approach......Page 285
4.1 Descent Problems and Their Scions......Page 289
4.2 Subclasses of Continuous Functions......Page 292
4.3 Local Tests and Reduction Steps......Page 294
4.4 Proof of the Main Algebraic Theorem......Page 298
4.5 Semialgebraic, Real, and p-Adic Analytic Cases......Page 302
References......Page 303
Recurrence for Stationary Group Actions......Page 305
1 Poincaré Recurrence for Stationary Actions......Page 307
2 Multiple Recurrence for SAT Actions......Page 308
3.1 Factors and the Disintegration of Measures......Page 310
4 Multiple Recurrence for Stationary Actions......Page 311
References......Page 313
On the Honda - Kaneko Congruences......Page 314
References......Page 322
1 Introduction......Page 324
2 Meromorphic Abelian Differentials of the Second Kind......Page 326
3 Meromorphic Abelian Differentials of the Third Kind......Page 332
4 Duality......Page 338
References......Page 345
1 Introduction......Page 346
2 Definitions and Preliminaries......Page 347
3 Main Results......Page 350
References......Page 355
1 A Formula for the Formal Dimension......Page 356
2 A Construction of a Basis......Page 360
References......Page 361
1 Introduction......Page 362
2 The Vector-Valued Modular Form (σ, C)......Page 365
3 Proof of the Main Theorems......Page 366
References......Page 373
1 Introduction......Page 374
References......Page 390
1 Introduction......Page 391
2 A Very Simple Proof......Page 392
3 Two Remarks......Page 393
References......Page 394
1 Introduction......Page 395
2 Upper Bounds......Page 398
3 A Density Criterion......Page 402
4 The Alternating Group Method......Page 406
5 Principal Homomorphisms......Page 409
6 SO(3)-Dense Groups......Page 413
7 Appendix by Y. William Yu......Page 416
References......Page 417
Two Embedding Theorems......Page 418
1 Introduction......Page 419
2 Real Embeddings......Page 423
3 Involutive Structures......Page 426
4 Underlying Complexes......Page 428
5 CR Functions......Page 431
6 CR Maps into CN......Page 433
7 CR Embeddings......Page 436
References......Page 447
1 Introduction......Page 449
2 Plancherel–Polya-Type Inequalities......Page 451
3 Cubature Formulas on Manifolds Which are Exact on Band-Limited Functions......Page 455
4 Cubature Formulas on Compact Manifolds Which are Exact on Variational Splines......Page 457
5 Positive Cubature Formulas on Compact Manifolds......Page 460
6 Harmonic Analysis on Compact Homogeneous Manifolds......Page 465
7 On the Product of Eigenfunctions of the Casimir Operator L on Compact Homogeneous Manifolds......Page 467
References......Page 470
The Moment Zeta Function and Applications......Page 472
1 A Formula for the Winning Time T......Page 476
2 Moment Zeta Function......Page 477
3 α> 1......Page 479
4 α= 1......Page 481
5 Riemann Zeta Function......Page 486
5.1 A Sum and an Integral......Page 488
References......Page 491
A Transcendence Criterion for CM on Some Familiesof Calabi–Yau Manifolds......Page 492
1 Introduction......Page 493
2 The Problem and the Main Results......Page 494
3 The Main Lemmas......Page 497
4 The Borcea Family as a Two-Step Tower......Page 498
5 The Viehweg–Zuo Family......Page 500
6 The First Step of a Borcea-Voisin Tower......Page 502
References......Page 506
1 Introduction......Page 508
2.1 Existence and Approximation of Solutions......Page 510
2.2 The State of Affairs in the Scalar Case ca 1954......Page 511
3.1 Simple Algebra in the General Case......Page 512
3.2 Analytic Sheaf Theory to the Rescue......Page 513
3.3 Estimates and Their Exploitation......Page 514
4.1 Noetherian Operators......Page 517
4.2 Extension with Bounds: Final Statement......Page 521
Reference......Page 524
1 Introduction......Page 525
2 The Flickering Circles......Page 526
3 Minimal Entire Functions......Page 528
4 Concluding Remarks......Page 530
References......Page 532
A Conjecture by Leon Ehrenpreis About Zeroes of Exponential Polynomials......Page 533
1 The Conjecture, Various Formulations......Page 534
2 What is Known in Connection with Results in Transcendental Number Theory......Page 537
3 Using the Formal Counterpart of Schanuel\'s Numerical Conjecture......Page 538
4.1 Lindemann–Weierstrass Theorem Versus Ritt\'s Factorization......Page 540
4.2 A First Ingredient for the Proof of Theorem 4.2: The Notion of ``Size\'\' for a Xd/dX Module over K(X)......Page 541
4.3 A Second Ingredient for the Proof of Theorem 4.2: A Theorem by N. Katz......Page 542
4.4 The D-Modules Approach......Page 543
References......Page 549
The Discrete Analog of the Malgrange–Ehrenpreis Theorem......Page 552
References......Page 556
The Legacy of Leon Ehrenpreis......Page 557
Phd Students of Leon Ehrenpreis......Page 558
Publications of Leon Ehrenpreis......Page 559