Geometric analysis on symmetric spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
1997
|
| Edition: | Reprinted with corrections |
| Series: | Mathematical surveys and monographs
Volume 39 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 611 Seiten Illustrationen, Diagramme |
| ISBN: | 0821815385 |
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MARC
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| 245 | 1 | 0 | |a Geometric analysis on symmetric spaces |c Sigurdur Helgason |
| 250 | |a Reprinted with corrections | ||
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 1997 | |
| 300 | |a xiv, 611 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v Volume 39 | |
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Record in the Search Index
| _version_ | 1804130222627356672 |
|---|---|
| adam_text | CONTENTS
Preface.
CHAPTER I. A Duality in Integral Geometry. 1
§1. Generalities. 4
1. Notation and Preliminaries 4
2. Principal Problems 6
§2. The Radon Transform for Points and Hyperplanes. 9
1. The Principal Results 9
2. The Kernel of the Dual Transform 15
3. The Radon Transform and its Dual on the if types 20
4. Inversion of the Dual Transform 23
5. The Range Characterization for Distributions and
Consequences 28
6. Some Facts about Topological Vector Spaces 32
§3. Homogeneous Spaces in Duality. 34
1. The Radon Transform for a Double Fibration 34
2. The Radon Transform for Grassmaimians 43
3. The Totally Geodesic Radon Transform on Constant
Curvature Spaces 50
A. The Generalized Transforms 50
B. The Inversion Formulas 51
C. Support Theorems 56
4. The d Plane Transform 59
5. Analogy with the Poisson Integral 64
Exercises and Further Results. 66
Notes. 72
vii
viii CONTENTS
CHAPTER II. A Duality for Symmetric Spaces. 75
§1. The Space of Horocycles. 76
1. Definition and Coset Representation 76
2. The Isotropy Actions for X and E 78
3. Geodesies in the Horocycle Space 82
§2. Invariant Differential Operators. 87
1. The Isomorphisms 87
2. Radial Part Interpretation 91
3. Joint Eigenspaces and Eigenspace Representations 93
4. The Mean Value Operators 94
§3. The Radon Transform and its Dual. 100
1. Measure theoretic Preliminaries 100
2. Integral Transforms and Differential Operators 103
3. The Inversion Formula and the Plancherel Formula
for the Radon Transform 108
4. The Poisson Transform 118
5. The Dual Transform and the Poisson Kernel 121
§4. Finite dimensional Spherical and Conical
Representations. 124
1. Conical Distributions. Elementary Properties 124
2. Conical Functions and Finite dimensional
Representations 132
3. The Finite dimensional Spherical Representations 139
4. Conical Models and Spherical Models 141
5. Simultaneous Euclidean Imbeddings of X and of E.
Horocycles as plane sections 143
6. Restricted Weights 148
7. The Component H(n) 151
§5. Conical Distributions. 154
1. The Construction of * A s 154
2. The Reduction to RankOne 158
3. The Analytic Continuation of ^ ,s 162
4. The Determination of the Conical Distributions 173
§6. Some Rank One Results. 179
1. Component Computations 179
2. The Inversion of W 182
3. The Simplicity Criterion 188
4. The Algebra B(K/M) 191
CONTENTS ix
5. An Additional Conical Distribution for A = 0 192
6. Conical Distributions for the Exceptional A 195
Exercises and Further Results. 205
Notes. 218
CHAPTER III. The Fourier Transform on a Symmetric Space. 221
§1. The Inversion and the Plancherel Formula. 222
1. The Symmetry of the Spherical Function 222
2. The Plancherel Formula 227
§2. Generalized Spherical Functions (Eisenstein Integrals). 233
1. Reduction to Zonal Spherical Functions 233
2. The Expansion of $a, 5 240
3. Simplicity (preliminary results) 248
§3. The Q* matrices. 250
1. The if finite functions in £A(5) 250
2. Connections with Harmonic Polynomials 252
3. A Product Formula for det(Q*(A)) (preliminary version) 256
§4. The Simplicity Criterion. 263
§5. The Paley Wiener Theorem for the Fourier Transform
on X = G/K. 269
1. Estimates of the r coefficients 270
2. Some Identities for Cs 273
3. The Fourier Transform and the Radon Transform.
iT types 275
4. Completion of the Proof of the Paley Wiener Theorem.
The Range £ (X)~ 278
5. A Topological Paley Wiener Theorem for the if types 283
6. The Inversion—and the Plancherel Formula for the
6 spherical Transform 290
§6. Eigenfunctions and Eigenspace Representations. 293
1. The if finite Eigenfunctions of B(X) 294
2. The Irreducibility Criterion for the Eigenspace Repre¬
sentations on G/K 295
§7. Tangent Space Analysis. 297
1. Discussion 297
2. The J polynomials 298
x CONTENTS
3. Generalized Bessel Functions and Zonal Spherical
Functions 304
4. The Fourier Transform of if finite Functions 306
5. The Range £ (p)~ inside H(a*x K/M) 312
§8. Eigenfunctions and Eigenspace Representations on Xo. 314
1. Simplicity 314
2. The .PC finite Joint Eigenfunctions of D(GO/K) 317
3. The Irreducibility Criterion for the Eigenspace Repre¬
sentations of Go/K 324
§9. The Compact Case. 325
1. Motivation 325
2. Compact Symmetric Spaces 327
3. Analogies 331
4. The Product Decomposition 332
§10. Elements of B(G/K) as Fractions. 338
§11. The Rank One Case. 343
1. An Explicit Formula for the Eisenstein Integral 343
2. Harmonic Analysis of K finite Functions 349
§12. The Spherical Transform Revisited. 352
1. Positive Definite Functions 352
2. The Spherical Transform for Gelfand Pairs 357
3. The Case of a Symmetric Space G/K 364
Exercises and Further Results. 370
Notes. 376
CHAPTER IV. The Radon Transform on X and on Xo.
Range Questions. 381
§1. The Support Theorem. 381
§2. The Ranges D(X)A and £ (X)A and £(H)V. 384
§3. The Range and Kernel for if types. 388
1. The General Case 388
2. Examples: H2 and R2 393
§4. The Radon Transform and its Dual for K invariants. 396
§5. The Radon Transform on Xo. 399
1. Preliminaries 399
CONTENTS xi
2. The Support Theorem 405
3. The Range and the Kernel for the if types 407
4. The Ranges £ {XO)A and £(EO)V 408
Exercises and Further Results. 411
Notes. 412
CHAPTER V. Differential Equations on Symmetric Spaces. 413
§1. Solvability. 413
1. Fundamental Solution of D 414
2. Solvability in £(X) 415
3. Solvability in £ {X) 419
§2. Mean Value Theorems. 420
1. The Mean Value Operators 420
2. Approximations by Analytic Functions 423
3. Asgeirsson s Mean Value Theorem extended to Homo¬
geneous Spaces. 425
§3. Harmonic Functions on Symmetric Spaces. 429
1. Generalities 429
2. Bounded Harmonic Functions 429
3. The Poisson Integral Formula for X 434
4. The Fatou Theorem 439
5. The Furstenberg Compactification 449
§4. Harmonic Functions on Bounded Symmetric Domains. 452
1. The Bounded Realization of a Hermitian Symmetric Space 452
2. The Geodesies in a Bounded Symmetric Domain 454
3. The Restricted Root Systems for Bounded Symmetric
Domains 455
4. The Action of Go on D and the Polydisk in D 462
5. The Shilov Boundary of a Bounded Symmetric Domain 464
6. The Dirichlet Problem for the Shilov Boundary 471
7. The Hua Equations 472
8. Integral Geometry Interpretation 478
§5. The Wave Equation on Symmetric Spaces. 480
1. Introduction. Huygens Principle 480
2. Huygens Principle for Compact Groups and Symmetric
Spaces X = G/K (G complex) 483
3. Huygens Principle and Cartan Subgroups 489
4. Orbital Integrals and Huygens Principle 494
5. Energy Equipartition 498
xii CONTENTS
§6. Eigenfunctions and Hyperfunctions. 503
1. Arbitrary Eigenfunctions 503
2. Exponentially Bounded Eigenfunctions 508
Exercises and Further Results. 509
Notes. 514
CHAPTER VI. Eigenspace Representations. 517
§1. Generalities. 517
1. A Motivating Example 517
2. Eigenspace Representations on Function and Distribution
Spaces 518
3. Eigenspace Representations for Vector Bundles 519
§2. Irreducibility Criteria for a Symmetric Space. 521
1. The Compact Case 521
2. The Euclidean Type 524
3. The Noncompact Type 524
§3. Eigenspace Representations for the Horocycle Space
G/MN. 526
1. The Principal Series 526
2. The Spherical Principal Series. Irreducibility 527
3. Conical Distributions and the Construction of the Inter¬
twining Operators 533
4. Convolution on G/MN 536
§4. Eigenspace Representations for the Complex Space G/N. 541
1. The Algebra D(G/JV) 541
2. The Principal,Series 544
3. The Finite Dimensional Holomorphic Representations 544
§5. Two Models of the Spherical Representations. 546
Exercises and Further Results. 549
Notes. 552
SOLUTIONS TO EXERCISES. 553
BIBLIOGRAPHY. 581
SYMBOLS FREQUENTLY USED. 603
INDEX. 607
|
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| author | Sigurður Helgason 1927-2023 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/62 |
| dewey-search | 516.3/62 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Reprinted with corrections |
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| id | DE-604.BV017402858 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:17:36Z |
| institution | BVB |
| isbn | 0821815385 |
| language | English |
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| physical | xiv, 611 Seiten Illustrationen, Diagramme |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | American Mathematical Society |
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| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Sigurður Helgason 1927-2023 Verfasser (DE-588)123045762 aut Geometric analysis on symmetric spaces Sigurdur Helgason Reprinted with corrections Providence, Rhode Island American Mathematical Society 1997 xiv, 611 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 39 Geometrische Analysis (DE-588)4156708-0 gnd rswk-swf Symmetrischer Raum (DE-588)4184206-6 gnd rswk-swf Symmetrischer Raum (DE-588)4184206-6 s Geometrische Analysis (DE-588)4156708-0 s DE-604 Gefolgt von 978-0-8218-4530-1 Mathematical surveys and monographs Volume 39 (DE-604)BV000018014 39 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010487020&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sigurður Helgason 1927-2023 Geometric analysis on symmetric spaces Mathematical surveys and monographs Geometrische Analysis (DE-588)4156708-0 gnd Symmetrischer Raum (DE-588)4184206-6 gnd |
| subject_GND | (DE-588)4156708-0 (DE-588)4184206-6 |
| title | Geometric analysis on symmetric spaces |
| title_auth | Geometric analysis on symmetric spaces |
| title_exact_search | Geometric analysis on symmetric spaces |
| title_full | Geometric analysis on symmetric spaces Sigurdur Helgason |
| title_fullStr | Geometric analysis on symmetric spaces Sigurdur Helgason |
| title_full_unstemmed | Geometric analysis on symmetric spaces Sigurdur Helgason |
| title_short | Geometric analysis on symmetric spaces |
| title_sort | geometric analysis on symmetric spaces |
| topic | Geometrische Analysis (DE-588)4156708-0 gnd Symmetrischer Raum (DE-588)4184206-6 gnd |
| topic_facet | Geometrische Analysis Symmetrischer Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010487020&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT sigurðurhelgason geometricanalysisonsymmetricspaces |