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Radon Transform 2nd ed. 1999 [Kietas viršelis]

  • Formatas: Hardback, 193 pages, aukštis x plotis: 235x155 mm, weight: 1050 g, XIII, 193 p., 1 Hardback
  • Serija: Progress in Mathematics 5
  • Išleidimo metai: 01-Aug-1999
  • Leidėjas: Birkhauser Boston Inc
  • ISBN-10: 0817641092
  • ISBN-13: 9780817641092
Kitos knygos pagal šią temą:
Radon Transform 2nd ed. 1999Didesnis paveikslėlis
  • Formatas: Hardback, 193 pages, aukštis x plotis: 235x155 mm, weight: 1050 g, XIII, 193 p., 1 Hardback
  • Serija: Progress in Mathematics 5
  • Išleidimo metai: 01-Aug-1999
  • Leidėjas: Birkhauser Boston Inc
  • ISBN-10: 0817641092
  • ISBN-13: 9780817641092
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780817641092.html
Raktažodžiai:
The Radon transform is an important topic in integral geometry which deals with the problem of expressing a function on a manifold in terms of its integrals over certain submanifolds. Solutions to such problems have a wide range of applications, namely to partial differential equations, group representations, X-ray technology, nuclear magnetic resonance scanning, and tomography. This second edition, significantly expanded and updated, presents new material taking into account some of the progress made in the field since 1980. Aimed at beginning graduate students, this monograph will be useful in the classroom or as a resource for self-study. Readers will find here an accessible introduction to Radon transform theory, an elegant topic in integral geometry.

The Radon transform is an important topic in integral geometry which deals with the problem of expressing a function on a manifold in terms of its integrals over certain submanifolds. Solutions to such problems have a wide range of applications, namely to partial differential equations, group representations, X-ray technology, nuclear magnetic resonance scanning, and tomography. This second edition, significantly expanded and updated, presents new material taking into account some of the progress made in the field since 1980.The first chapter introduces the Radon transform and presents new material on the d-plane transform and applications to the wave equation. Chapter 2 places the Radon transform in a general framework of integral geometry known as a double fibration of a homogeneous space. Several significant examples are developed in detail. Two subsequent chapters treat some specific examples of generalized Radon transforms, for examples, antipodal manifold in compact 2-points homogeneous spaces, and orbital integrals in isotropic Lorentzian manifolds. A final chapter deals with Fourier transforms and distributions, developing all the tools needed in the work.Aimed at beginning graduate students, this monograph will be useful in the classroom or as a resource for self-study. Readers will find here an accessible introduction to Radon transform theory, an elegant topic in integral geometry.

Recenzijos

"This is the second edition of the famous book by Sigurdur Helgason which has been updated in accordance with recent new results in this area. The list of references and bibliographical notes have been essentially extended. Many examples with explicit inversion formulas and range theorems have been added, and the group-theoretic viewpoint is emphasized... [ the] author adds a new chapter, Chapter 5, which contains useful information about Fourier transforms, distributions and Riesz potentials. The second edition preserves the nice introductory flavor of the first one. The book will be highly appreciated by the mathematical community."



--Mathematical Reviews (on the second edition)



"...well-written and a pleasure to read...provides clear explanations and illustrative figures... Every important point receives a clear proof. The author puts a great deal of effort into motivating his readers. The chapters have been updated by the inclusion of some applications and by giving indications in bibliographical notes of some recent developments... [ an] excellent introduction to an area of mathematics which seems to have attracted much new interest...highly recommendable...for graduate students... and researchers."



--ZAA (on the second edition)









"Until now the subject [ The Radon transform] has lacked anything approaching a systematic exposition aimed at beginners. Publication of the present volume...by one of the chief contributors to the modern theory of the Radon transform, is thustimely and welcome...Helgason's notes provide the most agreeable introduction to the Radon transform currently available.



[ A] reader will be charmed by the interplay of geometry and analysis exhibited here and reassured by the explicit nature of the formulas obtained. [ Chapter 3] is the heart of the second half of the book and, on balance, provides an admirable introit to a branch of analysis which deserves to be known by a wider public."



--SIAM Review (on the first edition)



 



 

Preface to the Second Edition ix(2)
Preface to the First Edition xi
CHAPTER I The Radon Transform on R^n
1(52)
1. Introduction
1(1)
2. The Radon Transform of the Spaces D(R^n) and S(R^n). The Support Theorem
2(13)
3. The Inversion Formula
15(5)
4. The Plancherel Formula
20(2)
5. Radon Transform of Distributions
22(6)
6. Integration over d-Planes. X-ray Transforms. The Range of the d-Plane Transform
28(13)
7. Applications
41(10)
a) Partial Differential Equations
41(5)
b) X-ray Reconstruction
46(5)
Bibliographical Notes
51(2)
CHAPTER II A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals
53(30)
1. Homogeneous Spaces in Duality
53(4)
2. The Radon Transform for the Double Fibration
57(5)
3. Orbital Integrals
62(1)
4. Examples of Radon Transforms for Homogeneous Spaces in Duality
63(17)
A. The Funk Transform
63(3)
B. The X-ray Transform in H^2
66(1)
C. The Horocycles in H^2
67(4)
D. The Poisson Integral as a Radon Transform
71(2)
E. The d-Plane Transform
73(1)
F. Grassmann Manifolds
74(1)
G. Half-lines in a Half-plane
75(4)
H. Theta Series and Cusp Forms
79(1)
Bibliographical Notes
80(3)
CHAPTER III The Radon Transform on Two-Point Homogeneous Spaces
83(40)
1. Spaces of Constant Curvature. Inversion and Support Theorems
83(28)
A. The Hyperbolic Space
85(8)
B. The Spheres and the Elliptic Spaces
93(15)
C. The Spherical Slice Transform
108(3)
2. Compact Two-Point Homogeneous Spaces. Applications
111(7)
3. Noncompact Two-Point Homogeneous Spaces
118(1)
4. The X-ray Transform on a Symmetric Space
119(1)
5. Maximal Tori and Minimal Spheres in Compact Symmetric Spaces
120(2)
Bibliographical Notes
122(1)
CHAPTER IV Orbital Integrals and the Wave Operator for Isotropic Lorentz Spaces
123(24)
1. Isotropic Spaces
123(5)
A. The Riemannian Case
124(1)
B. The General Pseudo-Riemannian Case
124(4)
C. The Lorentzian Case
128(1)
2. Orbital Integrals
128(9)
3. Generalized Riesz Potentials
137(3)
4. Determination of a Function from Its Integrals over Lorentzian Spheres
140(4)
5. Orbital Integrals and Huygens' Principle
144(1)
Bibliographical Notes
145(2)
CHAPTER V Fourier Transforms and Distributions. A Rapid Course
147(24)
1. The Topology of the Spaces D(R^n), SIGMA(R^n) and S(R^n)
147(2)
2. Distributions
149(1)
3. The Fourier Transform
150(6)
4. Differential Operators with Constant Coefficients
156(4)
5. Riesz Potentials
160(8)
Bibliographical Notes
168(3)
Bibliography 171(14)
Notational Conventions 185(2)
Subject Index 187