Integral geometry and radon transforms:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
2010
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIII, 301 S. graph. Darst. |
| ISBN: | 9781441960542 1441960546 9781441960559 |
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MARC
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| 245 | 1 | 0 | |a Integral geometry and radon transforms |c Sigurdur Helgason |
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| 300 | |a XIII, 301 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text |
Contents
Preface
vu
CHAPTER I
The Radon Transform on R™
§1
Introduction
1
§2
The Radon Transform of the Spaces
ЩИ71)
and <S(Rn). The
Support Theorem
2
§3
The Inversion Formula. Injectivity Questions
16
§4
The Plancherel Formula
25
§5
Radon Transform of Distributions
27
§6
Integration over d-planes. X-ray Transforms. The Range of the
cř-plane
Transform
32
§7
Applications
46
A. Partial Differential Equations. The Wave Equation
46
B. X-ray Reconstruction
52
Exercises and Further Results
56
Bibliographical Notes
60
CHAPTER II
A Duality in Integral Geometry
§1
Homogeneous Spaces in Duality
63
§2
The Radon Transform for the Double Fibration
67
(i) Principal Problems
68
(ii) Ranges and Kernels. General Features
71
(iii)
The Inversion Problem. General Remarks
72
§3
Orbital Integrals
75
§4
Examples of Radon Transforms for Homogeneous Spaces in Duality
77
A. The Funk Transform
77
B. The X-ray Transform in H2
80
C. The
Horocycles
in H2
82
D. The
Poisson
Integral as a Radon Transform
86
E. The ¿/-plane Transform
88
F. Grassmann
Manifolds
90
G. Half-lines in a Half-plane
91
H. Theta Series and Cusp Forms
94
I. The Plane-to-Line Transform in R}. The Range
95
J. Xoncompact Symmetric Space and Its Family of
Horocycles
103
Exercises and Further Results
104
Bibliographical
Xotes
108
Xli
CHAPTER III
The Radon Transform on Two-Point Homogeneous Spaces
§1
Spaces of Constant Curvature. Inversion and Support Theorems 111
A. The Euclidean Case R™
114
B. The Hyperbolic Space
118
C. The Spheres and the Elliptic Spaces
133
D. The Spherical Slice Transform
145
§2
Compact Two-Point Homogeneous Spaces. Applications
147
§3
Noncompact
Τ
wo-Point Homogeneous Spaces
157
§4
Support Theorems Relative to
Horocycles
159
Exercises and Further Results
167
Bibliographical Notes
168
CHAPTER IV
The
Х
-Ray Transform on a Symmetric Space
§1
Compact Symmetric Spaces. Injectivity and Local Inversion.
Support Theorem
171
§2
Noncompact Symmetric Spaces. Global Inversion and General
Support Theorem
178
§3
Maximal Tori and Minimal Spheres in Compact Symmetric Spaces
180
Exercises and Further Results
182
Bibliographical Notes
183
CHAPTER V
Orbital Integrals and the Wave Operator for
Isotropie
Lorentz
Spaces
§1 Isotropie
Spaces
185
A. The Riemannian Case
186
B. The General
Pseudo-
Riemannian Case
186
C. The Lorentzian Case
190
§2
Orbital Integrals
190
§3
Generalized Riesz Potentials
199
§4
Determination of a Function from Its Integral over Lorentzian
Spheres
202
§5
Orbital Integrals and Huygens' Principle
206
Bibliographical Notes
208
CHAPTER VI
The Mean-Value Operator
§1
An Injectivity Result
209
§2
Asgehssoivs Mean-Value Theorem Generalized
211
§3
John's Identities
215
Exercises and Further Results
217
Bibliographical Notes
219
хш
CHAPTER
VU
Fourier Transforms and Distributions. A Rapid Course
§1
The Topology of Spaces X>(Rn), £(Rn), and S(Rn)
221
§2
Distributions
223
§3
Convolutions
224
§4
The Fourier Transform
226
§5
Differential Operators with Constant Coefficients
234
§6
Riesz Potentials
236
Exercises and Further Results
248
Bibliographical Notes
250
CHAPTER
VIII
Lie Transformation Groups and Differential Operators
§1
Manifolds and Lie Groups
253
§2
Lie Transformation Groups and Radon Transforms
261
CHAPTER IX
Symmetric Spaces
§1
Definition and Examples
265
§2
Symmetric Spaces of the Noncompact Type
267
§3
Symmetric Spaces of the Compact Type
273
Bibliography
275
Notational Conventions
295
Frequently Used Symbols
297
Index
299 |
| any_adam_object | 1 |
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| building | Verbundindex |
| bvnumber | BV037314613 |
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| ctrlnum | (OCoLC)698550158 (DE-599)DNB1008069051 |
| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-20T11:04:11Z |
| institution | BVB |
| isbn | 9781441960542 1441960546 9781441960559 |
| language | English |
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| physical | XIII, 301 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| spelling | Sigurður Helgason 1927-2023 (DE-588)123045762 aut Integral geometry and radon transforms Sigurdur Helgason New York, NY [u.a.] Springer 2010 XIII, 301 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Integralgeometrie (DE-588)4161911-0 gnd rswk-swf Radon-Transformation (DE-588)4479199-9 gnd rswk-swf Integralgeometrie (DE-588)4161911-0 s Radon-Transformation (DE-588)4479199-9 s DE-604 Erscheint auch als online-Ausgabe 978-1-4419-6055-9 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3556925&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022468949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sigurður Helgason 1927-2023 Integral geometry and radon transforms Integralgeometrie (DE-588)4161911-0 gnd Radon-Transformation (DE-588)4479199-9 gnd |
| subject_GND | (DE-588)4161911-0 (DE-588)4479199-9 |
| title | Integral geometry and radon transforms |
| title_auth | Integral geometry and radon transforms |
| title_exact_search | Integral geometry and radon transforms |
| title_full | Integral geometry and radon transforms Sigurdur Helgason |
| title_fullStr | Integral geometry and radon transforms Sigurdur Helgason |
| title_full_unstemmed | Integral geometry and radon transforms Sigurdur Helgason |
| title_short | Integral geometry and radon transforms |
| title_sort | integral geometry and radon transforms |
| topic | Integralgeometrie (DE-588)4161911-0 gnd Radon-Transformation (DE-588)4479199-9 gnd |
| topic_facet | Integralgeometrie Radon-Transformation |
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