Selected topics in integral geometry:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Providence, RI
American Math. Soc.
2003
|
| Series: | Translations of mathematical monographs
220 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010351956&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | IX, 170 S. |
| ISBN: | 0821829327 |
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MARC
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| 240 | 1 | 0 | |a Izbrannye zadači integral'noj geometrii |
| 245 | 1 | 0 | |a Selected topics in integral geometry |c I. M. Gelfand ; S. G. Gindikin ; M. I. Graev |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2003 | |
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Record in the Search Index
| _version_ | 1819244899056222208 |
|---|---|
| adam_text | Contents
Preface to the English Edition xi
Preface xiij
Chapter 1. Radon Transform 1
1. Radon transform on the plane 1
1.1. Radon transform on the Euclidean plane 1
1.2. Inversion formula 2
1.3. Remarks 4
1.4. Radon transform on the affine plane 4
1.5. Relation to the Fourier transform and another proof of the inversion
formula 5
2. Radon transform in three dimensional space 7
2.1. Radon transform in Euclidean space 7
2.2. Radon transform in the affine space 9
2.3. Radon transform for a space of arbitrary dimension 10
3. Wave equation and the Huygens principle 12
3.1. Two dimensional case 12
3.2. Three dimensional case 13
4. Cavalieri s conditions and Paley Wiener theorems
for the Radon transform 14
4.1. Cavalieri s conditions for rapidly decreasing functions 14
4.2. Paley Wiener theorem for the space S(]R2) 15
4.3. Paley Wiener theorem for the space P(R2) of compactly supported
infinitely differentiable functions 16
4.4. Inversion of the Radon transform of a function / 6 X (M2) using the
moments 17
4.5. Reconstruction of unknown directions from known values of TZ f 17
5. Poisson formula for the Radon transform, and the discrete Radon
transform 19
5.1. Poisson formula for the Radon transform on a plane 19
5.2. Discrete Radon transform; relation to the Fourier series 21
5.3. Problem of integral geometry on the torus 21
6. Minkowski Funk transform 22
7. Radon transform of differential forms 25
7.1. Radon transform of 1 forms on the plane 25
7.2. Radon transform of 2 forms on the plane 26
7.3. Radon transform of 2 forms in three dimensional space 28
7.4. Radon transform of 3 forms in three dimensional space 29
8. Radon transform for the projective plane and projective space 30
V
vi CONTENTS
8.1. Spaces P3 and (P3) 30
8.2. Radon transform for P3 31
8.3. Relation to the affirie Radon transform for R3 and to the
Minkowski Funk transform for the three dimensional sphere 32
8.4. Inversion formula for the Radon transform on P3 32
8.5. On the inversion formulas for the afnne Radon transform on K3 and
the Minkowski Funk transform for S3 34
8.6. Description of the image of the Radon transform for P3 35
8.7. Radon transform for the projective plane P2 35
8.8. Radon transform for the projective space of an arbitrary dimension 38
9. Radon transform on the complex afnne space 38
9.1. Definition of the Radon transform 39
9.2. Relation to the Fourier transform 39
9.3. Inversion formula for the Radon transform 40
9.4. Case n = 2 40
9.5. Relation to Paley Wiener theorems for the afnne Radon transform
in R2 and R3 40
Chapter 2. John Transform 43
1. John transform in the real afnne space 44
1.1. John transform in R3 44
1.2. John transform and the Gauss hypergeometric function 45
1.3. Theorem on the image of the operator J 46
1.4. Space S(H ) 48
1.5. Description of the image of S(R3) in the space S(H ) 50
1.6. Proof of Theorem 1.1 on the image of the John transform 50
1.7. Analogs of the operator k 51
2. John transform of differential forms on R3 53
2.1. Definition of the John transform of differential forms 53
2.2. John transform of 3 forms on K3 54
2.3. John transform of 2 forms on R3 56
2.4. John transform of 1 forms on R3 58
3. John transform in the three dimensional real projective space 59
3.1. Manifold of lines in P3 59
3.2. John transform in P3 60
3.3. Relation to the John transform in the affine space 61
3.4. Description of the image of the John transform 62
3.5. Another way to define the John transform 63
3.6. Proof of the theorem on the image of the John transform 64
3.7. John transform as an intertwining operator 65
4. John transform in the complex affine space 67
4.1. John transform in C3 67
4.2. Differential form Kip and the theorem on the image of the John
transform 68
4.3. Inversion formula 68
4.4. Analogs of the operator k 69
5. Problems of integral geometry for line complexes in C3 71
5.1. Problem of integral geometry for a complex of lines in C3 intersecting
a curve 71
CONTENTS vii
5.2. Definition of admissible line complexes in C3 72
5.3. Necessary and sufficient conditions for a complex K to be admissible 73
5.4. Geometric structure of admissible complexes 75
5.5. Description of admissible complexes 76
Chapter 3. Integral Geometry and Harmonic Analysis on the Hyperbolic
Plane and in the Hyperbolic Space 79
1. Elements of hyperbolic planimetry 79
1.1. Models of the hyperbolic plane 79
1.2. Horocycles 81
1.3. Geodesies 82
2. Horocycle transform 83
2.1. Definition of the operator Kh 83
2.2. Inversion formula 83
2.3. Asgeirsson relations 85
2.4. Symmetry relation 86
2.5. Inversion formula for the horocycle transform in another model of
the hyperbolic plane 86
3. Analog of the Fourier transform on the hyperbolic plane and the relation
between this analog and the horocycle transform 86
3.1. Fourier transform on R2 86
3.2. Fourier transform on the hyperbolic plane 88
3.3. Relation to the horocycle transform and the inversion formula 88
3.4. Symmetry relation 90
3.5. Plancherel formula 91
4. Relation to the representation theory of the group SL(2,R) 91
5. Integral transform related to lines (geodesies)
on the hyperbolic plane C? 93
5.1. Definition and the inversion formula in the Poincare model 93
5.2. Relation to the Radon transform on the projective plane 95
6. Horospherical transform in the three dimensional
hyperbolic space £3 96
6.1. Models of the hyperbolic space 96
6.2. Horospheres 97
6.3. Horospherical transform 98
6.4. Inversion formula 99
6.5. Symmetry relation 101
6.6. Inversion formula for the horospherical transform in another model
of the hyperbolic space 101
6.7. Integral transform related to completely geodesic surfaces in C? 102
7. Analog of the Fourier transform in the hyperbolic space,
and its relation to the horospherical transform 103
7.1. Definition of the Fourier transform 103
7.2. Inversion formula 104
7.3. Symmetry relation and the Plancherel formula 105
8. Relation to the representation theory for the group SL(2,C) 105
9. Wave equation for the hyperbolic plane and hyperbolic space, and the
Huygens principle 106
9.1. Two dimensional case 106
viii CONTENTS
9.2. Three dimensional case 108
Chapter 4. Integral Geometry and Harmonic Analysis
on the Group G = SL(2, C) 111
1. Geometry on the group G 111
1.1. Group G as a homogeneous space 111
1.2. Plane sections of the hyperboloid G 112
1.3. Manifold of horospheres 113
1.4. Embedding the manifold of horospheres H in the projective space 116
1.5. Line complex in C3 associated with the manifold of horospheres 117
1.6. Manifold of paraboloids 117
2. Integral geometry on the group G = SL(2, C) 119
2.1. Integral transforms related to the space H of horospheres and the
complex of lines K 119
2.2. Symmetry relations for the horospherical transform 121
2.3. Inversion formula for the integral transform IZo related to the line
complex XinC3 121
2.4. Inversion formula for the horospherical transform 123
2.5. Inversion formula for the horospherical transform on the hyperbolic
space £3 124
2.6. Integral transform related to paraboloids on G 125
3. Harmonic analysis on the group G = SL(2, C) 128
3.1. Laplace Beltrami operator on the group G 128
3.2. Horospherical functions on G 129
3.3. Fourier transform on G 131
3.4. Relation between the Fourier transform on G and the horospherical
transform 132
3.5. Symmetry relation for the Fourier transform 133
3.6. Inversion formula for the Fourier transform 133
3.7. Analog of the Plancherel formula 135
3.8. Relation between the Fourier transform on G and the representations
of the group G x G 136
3.9. Relation to the representations of the group G 137
4. Another version of the Fourier transform on G = SL(2,C) 139
4.1. Functions *x{g;£, Q 140
4.2. Fourier transform on G 141
4.3. Relation between the above two versions of the Fourier transform 141
4.4. Symmetry relation 142
4.5. Inversion formula and Plancherel formula for the Fourier transform JF142
4.6. Relations with representation theory 143
Chapter 5. Integral Geometry on Quadrics 145
1. Integral transform related to the hyperplane sections
of a hyperboloid of two sheets in Rn+1 145
1.1. Definition 145
1.2. Admissible submanifolds in the manifold of hyperplane sections of
Cn 147
1.3. Operator kx 148
1.4. Local and nonlocal operators n 151
CONTENTS ix
1.5. Inversion formula 152
1.6. Examples 153
2. Integral transform related to spheres in Euclidean space En 156
2.1. Definition 157
2.2. Operator kx 157
2.3. Inversion formula 159
2.4. Examples 159
Bibliography 165
Index 167
|
| any_adam_object | 1 |
| author | Gelʹfand, Izrailʹ M. 1913-2009 Gindikin, Semen G. 1937- Graev, Mark I. |
| author_GND | (DE-588)118831364 (DE-588)123010950 |
| author_facet | Gelʹfand, Izrailʹ M. 1913-2009 Gindikin, Semen G. 1937- Graev, Mark I. |
| author_role | aut aut aut |
| author_sort | Gelʹfand, Izrailʹ M. 1913-2009 |
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| classification_rvk | SK 370 SK 450 |
| ctrlnum | (OCoLC)52288750 (DE-599)BVBBV017173003 |
| dewey-full | 516.3/62 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/62 516.362 |
| dewey-search | 516.3/62 516.362 |
| dewey-sort | 3516.3 262 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV017173003 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T11:15:44Z |
| institution | BVB |
| isbn | 0821829327 |
| language | English Russian |
| lccn | 2003052222 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010351956 |
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| physical | IX, 170 S. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spellingShingle | Gelʹfand, Izrailʹ M. 1913-2009 Gindikin, Semen G. 1937- Graev, Mark I. Selected topics in integral geometry Translations of mathematical monographs Géométrie intégrale Integral geometry Integralgeometrie (DE-588)4161911-0 gnd |
| subject_GND | (DE-588)4161911-0 |
| title | Selected topics in integral geometry |
| title_alt | Izbrannye zadači integral'noj geometrii |
| title_auth | Selected topics in integral geometry |
| title_exact_search | Selected topics in integral geometry |
| title_full | Selected topics in integral geometry I. M. Gelfand ; S. G. Gindikin ; M. I. Graev |
| title_fullStr | Selected topics in integral geometry I. M. Gelfand ; S. G. Gindikin ; M. I. Graev |
| title_full_unstemmed | Selected topics in integral geometry I. M. Gelfand ; S. G. Gindikin ; M. I. Graev |
| title_short | Selected topics in integral geometry |
| title_sort | selected topics in integral geometry |
| topic | Géométrie intégrale Integral geometry Integralgeometrie (DE-588)4161911-0 gnd |
| topic_facet | Géométrie intégrale Integral geometry Integralgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010351956&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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