Offbeat Integral Geometry on Symmetric Spaces

Cover......Page 1
Offbeat Integral Geometry on Symmetric Spaces......Page 4
Contents......Page 6
Preface......Page 10
1.1 Notation......Page 12
1.2 Distributions......Page 18
1.3 Some transcendental functions......Page 27
1.4 Spherical harmonics......Page 36
1.5 The Gegenbauer polynomials......Page 43
1. The Titchmarsh theorem generalized (Voronin [V67])\r......Page 47
2. Discrete harmonics (Delsarte [D7])......Page 48
3. Spherical codes (Seidel [S10])......Page 49
Bibliographical notes......Page 50
2.1 Homeomorphisms with the generalized transmutation property......Page 52
2.2 Some completeness results......Page 62
2.3 Systems of convolution equations......Page 66
2.4 Abel type integral equations......Page 73
2. Fractional integral operators (Koornwinder [K10])......Page 87
4. A nonlinear Abel equation (Gorenflo and Vessella [G12])\r......Page 89
5. Support properties of Radon transforms on curves (Quinto [Q2])\r......Page 90
Bibliographical notes......Page 91
3.1 Generalities......Page 93
3.2 The mapping......Page 100
3.3 Uniqueness theorems......Page 109
1. The Jacobi transform......Page 111
2. The Abel transform......Page 113
3. The Jacobi convolution......Page 115
5. Characterization of the class E (X)......Page 116
Bibliographical notes......Page 117
4.1 Introductory considerations......Page 119
4.2 The functions Φλ,n,k,m,j......Page 125
4.3 Generalized spherical transform......Page 129
4.4 The mapping k,m,j\r......Page 135
2. Characterizations of the irreducibles......Page 139
Bibliographical notes......Page 141
5.1 Preliminaries......Page 143
5.2 The functions φλ,n,p,q,l......Page 146
5.3 The transform Fl(p,q)......Page 151
5.4 The mapping......Page 156
1. The Heisenberg group......Page 158
3. The Fourier transform on Hn (Geller [G6], Thangavelu [T3])\r......Page 159
4. The Laguerre convolution (McCully [M7], Askey [A14],\rGorlich-Markett [G13], Thangavelu [T2])......Page 160
5. Hermite functions and the twisted convolution (Folland [F7] and Thangavelu [T2])......Page 161
6. The Abel transform on Damek–Ricci spaces (Rouvière [R11], Peyerimhoff-Samiou [P2])......Page 162
Bibliographical notes......Page 164
1.1 Simplest properties of functions with zero integrals over balls......Page 165
1.2 Uniqueness results......Page 171
1.3 Description of functions in the classes Vr(BR) and Ur(BR)\r......Page 178
1.4 Local two-radii theorems......Page 184
1.5 Functions with zero integrals over balls in a spherical annulus......Page 203
1.6 The Liouville property......Page 207
1. Continuation of functions with vanishing integrals over balls (Zaraisky [Z9])......Page 229
3. Decomposition of vector fields (Smith [S22])\r......Page 230
5. Pompeiu transforms with real analytic weights......Page 231
8. Liouville property (Hansen [H1])......Page 232
Bibliographical notes......Page 233
2.1 Auxiliary constructions......Page 235
2.2 The Jacobi functions......Page 239
2.3 The operator A......Page 242
2.4 Functions with vanishing averages over geodesic balls......Page 247
2.5 A definitive version of the local two-radii theorem......Page 254
2.6 A local two-radii theorem for weighted ball means......Page 258
2.7 The compact case......Page 261
1. The local two-radii theorem on the Jacobi hypergroup (Selmi and Nessibi [S11])......Page 265
2. “Freak theorems” (Berenstein and Zalcman [B12])......Page 266
4. Mean value extension result (Quinto [Q4])......Page 267
5. Behavior at infinity......Page 268
Bibliographical notes......Page 269
3.1 The Berenstein–Gay–Yger result......Page 271
3.2 The Berenstein–Gay–Yger theorem generalized......Page 278
3.3 The case of a ball and a sphere......Page 287
A. One-radius theorem on two-point homogeneous spaces......Page 301
B. Over-determined interpolation problems......Page 306
2. Deconvolution problem (Hörmander [H11])......Page 309
5. Inversion of the antipodal map (Helgason [H7])......Page 310
6. The Kotelnikov–Shannon interpolation formula......Page 311
Bibliographical notes......Page 312
4.1 The Pompeiu problem......Page 314
4.2 Some examples of Pompeiu sets......Page 318
4.3 A characterization of Pompeiu sets......Page 329
4.4 The local Pompeiu property......Page 332
4.5 Upper and lower estimates for R(A)......Page 337
4.6 The value of R(A) for some subsets of the plane......Page 347
1. The Pompeiu property (Dalmasso [D3])......Page 358
2. The value of R(A) for other sets\r......Page 359
3. The local Pompeiu problem for hyperbolic polygons......Page 360
Bibliographical notes......Page 361
5.1 The value R(A) for convex polytopes......Page 363
5.2 The value R(A) for rectangular parallelepipeds......Page 369
5.3 The class (A, Br)......Page 375
1. The Pompeiu transform (Berenstein–Gay–Yger [B20])......Page 389
2. A local inverse (Volchkova [V66], Berenstein–Gay–Yger [B20])\r......Page 390
3. Integrals over translations of parallelepipeds (V.V. Volchkov [V5], [IG])......Page 391
4. Zalcman’s problem......Page 392
5. Mean value property (Iwasaki-Kenma-Matsumoto [I4])......Page 393
6. Mean value characterization of harmonic polynomials (V.V. Volchkov [IG])......Page 394
Bibliographical notes......Page 395
6.1 Requisite results......Page 396
6.2 Functions with zero averages over ellipsoids......Page 400
6.3 The value of R(A) for ellipsoids......Page 409
6.4 Reconstruction of a function by means of its integrals over ellipsoids of revolution......Page 420
6.5 Mean-value characterization of pluriharmonic and separately harmonic functions......Page 425
2. Howard’s identity (John [J3])......Page 432
4. Liouville’s property (Volchkov [V9])......Page 433
6. Characterization of null quadrature domains (Friedman-Sakai [F8])......Page 434
Bibliographical notes......Page 435
7.1 Auxiliary assertions......Page 437
7.2 The functions Uv,k and Ψk,l ν......Page 445
7.3 Basic properties of the class Vr (BR)......Page 449
7.4 Two-radii theorems for the class Vr (BR)......Page 454
A. Conical injectivity sets of the spherical Radon transform......Page 459
B. Partial Differential Equations......Page 464
7.6 The hemispherical transform......Page 466
7.7 Measures with the Pompeiu property......Page 473
7.8 The Pompeiu property for spherical polygons......Page 480
7.9 Extremal versions of the Pompeiu problem on a sphere......Page 485
1. Uniqueness theorem (V.V. Volchkov [V48])......Page 493
5. Characterization of harmonic and analytic functions......Page 494
6. Strip problem (Tumanov [T13])......Page 495
7. Morera theorems via microlocal analysis (Globevnik-Quinto [G9])......Page 496
Bibliographical notes......Page 497
8.1 Main problems......Page 499
8.2 Pompeiu transforms for distributions with support on a sphere......Page 503
8.3 The Pompeiu problem for groups......Page 513
8.4 Spherical means on the reduced Heisenberg group and the Pompeiu problem with a twist......Page 517
8.5 Pompeiu’s problem on discrete space......Page 520
1. Injectivity sets of PF (V.V. Volchkov [IG])\r......Page 523
2. The local two-radii theorem on the Chéble–Trimèche hypergroup (Selmi and Nessibi [S11], Trimèche [T12])......Page 524
3. Pompeiu transforms with real analytic weights (Quinto [Q1])......Page 525
6. Pompeiu sets on symmetric spaces......Page 526
Bibliographical notes......Page 527
9.1 Pompeiu transforms on a complete Riemannian manifold......Page 529
9.2 Radial Pompeiu transforms on a locally symmetric space......Page 533
9.3 Freak theorems......Page 537
9.4 General Pompeiu transforms on locally symmetric spaces......Page 542
3. Support theorem for the sphere transform (Quinto [Q4])......Page 555
Bibliographical notes......Page 556
Bibliography......Page 558
Author Index......Page 580
Subject Index......Page 585
Basic Notation......Page 588