Introduction to radon transforms: with elements of fractional calculus and harmonic analysis
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2015
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications
160 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and indexes Portion of title: Radon transforms |
| Beschreibung: | xvii, 576 Seiten Diagramme |
| ISBN: | 9780521854597 |
Internformat
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Datensatz im Suchindex
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| adam_text | Titel: Introduction to Radon transforms
Autor: Rubin, Boris
Jahr: 2015
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Introduction to Radon Transforms With Elements of Fractional Calculus and Harmonic Analysis BORIS RUBIN Louisiana State University Cambridge UNIVERSITY PRESS
Contents Preface page xi Notation and Conventions xv 1 Preliminaries 1 1.1 Basic Integral Inequalities 1 1.2 Integral Operators with Homogeneous Kernel 1 1.3 Analyticity of Functions Represented by Integrals 4 1.4 Gamma and Beta Functions 5 1.5 Finite Differences 6 1.6 Jacobi, Gegenbauer, and Chebyshev Polynomials 7 1.7 The Laplace Operator 10 1.8 The Fourier Transform on R 11 1.9 Operator Families and Approximate Identities 14 1.10 Distributions 19 1.11 Differentiable Manifolds and Mappings 23 1.12 Polar Coordinates and Integration on the Sphere 26 1.13 Some Useful Integrals 32 1.14 Notes to Chapter 1 33 2 Fractional Integration: Functions of One Variable 35 2.1 Fractional Integrals and Derivatives on a Finite Interval 35 2.2 Fractional Integrals on the Half-Line 39 2.3 Decreasing Fractional Integrals and Wavelet Measures 45 2.4 Fractional Derivatives on the Half-Line 52 2.5 Regularization of Integrals with Power Singularity and Analytic Continuation of Fractional Integrals 61 2.6 Some Other Fractional Integrals 64 2.7 Notes to Chapter 2 81 3 Riesz Potentials 83 3.1 Definition and Basic Properties 83 3.2 The Fourier Transform of Riesz Potentials 87 vii
Contents vi i i 3.3 The Scmyanistyi-Lizorkin Spaces 90 3.4 Factorization of Riesz Potentials 97 3.5 Inversion of Riesz Potentials 100 3.6 Notes to Chapter 3 125 4 The Radon Transform on R 127 4.1 Definitions and Elementary Properties 127 4.2 The Dual Radon Transform 138 4.3 Radon Transforms of Radial Functions. Connection with Fractional Integrals 140 4.4 Mapping Properties of the Radon Transform 144 4.5 Connection with Riesz Potentials 155 4.6 Connection with the Fourier Transform and Action on Smooth Functions 156 4.7 Mean Value Operators and Shifted Radon Transforms 173 4.8 Inversion of the Radon Transform 175 4.9 Semyanistyi s Fractional Integrals 205 4.10 Kolmogorov’s Problem for the Half-Space Integrals 214 4.11 Radon Transform of a Finite Measure 223 4.12 Radon Transforms and Spherical Harmonics 226 4.13 The Transversal Radon Transform 245 4.14 Radon Transform on the Heisenberg Group 268 4.15 Notes to Chapter 4 273 5 Operators of Integral Geometry on the Unit Sphere 280 5.1 Funk, Cosine, and Sine Transforms 280 5.2 Connection between the Funk Transform and the Radon Transform. Kernel and Support Theorems 316 5.3 The Hemispherical Transform and the Cosine Transform with Odd Kernel 320 5.4 The Buscmann-Petty Problem for Convex Bodies 343 5.5 Spherical Caps, Small Divisors, and Related Analytic Families 355 5.6 Notes to Chapter 5 361 6 Operators of Integral Geometry in the Hyperbolic Space 364 6.1 Preliminaries 364 6.2 The Totally Geodesic Radon Transform 388 6.3 The Horospherical Transform 416 6.4 Notes to Chapter 6 443 7 Spherical Mean Radon Transforms 445 7.1 Spheres through the Origin 445 7.2 The Spherical Slice Transform on S 448 7.3 Spherical Means with Centers on a Sphere 454 7.4 Notes to Chapter 7 466 A Elements of Harmonic Analysis on the Unit Sphere in R 467 A.l Introduction 467
Contents ix A. 2 The Space P„, of Homogeneous Polynomials 468 A. 3 Spherical Harmonics 470 A. 4 The Fouricr-Laplace Series 473 A. 5 Zonal Harmonics 478 A. 6 Spherical Polynomials 480 A. 7 The Funk-Hecke Formula 485 A. 8 Estimates of Spherical Harmonics and Their Derivatives 487 A.9 Relationship between the Smoothness of Functions and Convergence of Their Fourier-Laplace Series 490 A. 10 Generalized Functions on the Sphere 496 A. 11 Spherical Convolution Operators 499 A. 12 The Fourier-Faplace Multipliers 510 A. 13 Examples of Spherical Convolutions; Sobolev Spaces 522 A. 14 Maximal Functions and Approximate Identities 533 A. 15 Notes to Appendix A 537 B Open Problems 539 Bibliography 541 Subject Index 569 Index of Symbols 571 Author Index 573
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| isbn | 9780521854597 |
| language | English |
| lccn | 015012673 |
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| physical | xvii, 576 Seiten Diagramme |
| publishDate | 2015 |
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| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Rubin, Boris 1945- Verfasser (DE-588)1082771988 aut Introduction to radon transforms with elements of fractional calculus and harmonic analysis Boris Rubin, Louisiana State University Radon transforms Cambridge Cambridge University Press 2015 xvii, 576 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 160 Includes bibliographical references and indexes Portion of title: Radon transforms Integral geometry Radon-Transformation (DE-588)4479199-9 gnd rswk-swf Radon-Transformation (DE-588)4479199-9 s DE-604 Encyclopedia of mathematics and its applications 160 (DE-604)BV000903719 160 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028886751&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rubin, Boris 1945- Introduction to radon transforms with elements of fractional calculus and harmonic analysis Encyclopedia of mathematics and its applications Radon transforms Integral geometry Radon-Transformation (DE-588)4479199-9 gnd |
| subject_GND | (DE-588)4479199-9 |
| title | Introduction to radon transforms with elements of fractional calculus and harmonic analysis |
| title_alt | Radon transforms |
| title_auth | Introduction to radon transforms with elements of fractional calculus and harmonic analysis |
| title_exact_search | Introduction to radon transforms with elements of fractional calculus and harmonic analysis |
| title_full | Introduction to radon transforms with elements of fractional calculus and harmonic analysis Boris Rubin, Louisiana State University |
| title_fullStr | Introduction to radon transforms with elements of fractional calculus and harmonic analysis Boris Rubin, Louisiana State University |
| title_full_unstemmed | Introduction to radon transforms with elements of fractional calculus and harmonic analysis Boris Rubin, Louisiana State University |
| title_short | Introduction to radon transforms |
| title_sort | introduction to radon transforms with elements of fractional calculus and harmonic analysis |
| title_sub | with elements of fractional calculus and harmonic analysis |
| topic | Radon transforms Integral geometry Radon-Transformation (DE-588)4479199-9 gnd |
| topic_facet | Radon transforms Integral geometry Radon-Transformation |
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