Integral Geometry and Radon Transforms

ISBN-13: 9781489994202 / Angielski / Miękka / 2014 / 301 str.

Sigurdur Helgason

Opis

In this text, integral geometry deals with Radon's problem of representing a function on a manifold in terms of its integrals over certain submanifolds--hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial di erential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation. From the reviews: "Integral Geometry is a fascinating area, where numerous branches of mathematics meet together. the contents of the book is concentrated around the duality and double vibration, which is realized through the masterful treatment of a variety of examples. the book is written by an expert, who has made fundamental contributions to the area." --Boris Rubin, Louisiana State University

Recenzje

From the reviews:

"Beginning graduate students and interested nonspecialists will gain from the book because of its clear exposition and comprehensive nature. Practitioners of integral geometry will find it a valuable reference with complete and clear proofs as well as specialized items of interest ... . This book is a well-written and beautiful introduction to integral geometry from the perspective of group actions. It has valuable thought provoking exercises. ... It should be read by anyone who would like to learn more about integral geometry." (Fulton Gonzalez and Eric Todd Quinto, Bulletin of the American Mathematical Society, Vol. 50 (4), October, 2013)

"This specialist book in the field of geometric analysis is intended for graduate students and professionals. ... The field has several applications including medical X-ray technology and tomography. The nine-chapter volume devotes a chapter to these technologies and other applications. The book is largely self-contained, and includes a chapter at the end with necessary background material ... . The volume includes a substantial bibliography, and each chapter concludes with bibliographic notes. Summing Up: Recommended. Graduate students and above." (D. Z. Spicer, Choice, Vol. 48 (11), August, 2011)

"Integral Geometry and Radon Transforms is truly a pedagogical contribution by a master in this field, aiming at providing proper background - and then some - for entry into real work on the topics he brings into the spotlight. ... every page is filled with serious mathematics, and Helgason provides a lot of commentary and references that ought to be pursued by those wishing to enter the field. It is a marvelous example of fine scholarship." (Michael Berg, The Mathematical Association of America, January, 2011)

"The book under review is devoted to integral operators of geometric nature. The monograph deals with the problem of representing a function on a manifold in terms of its integrals over certain submanifolds - hence the term Radon transform. ... The book will be of interest for both students and specialists in integral geometry, analysis on homogeneous spaces and partial differential equations. It may serve as a source for further research in the area." (Valery Vladimirovich Volchkov, Zentralblatt MATH, Vol. 1210, 2011)

Spis treści

1.- The Radon Transformon on Rⁿ 1.1- Introduction 1.2- The Radon Transform: The Support Theorem 1.3- The Inversion Formula: Injectivity Questions 1.4- The Plancherel Formula 1.5- Radon Transform of Distribution 1.6- Integration over d-planes: X-Ray Transforms 1.7- Applications 2.- A Duality in Integral Geometry 2.1- Homogeneous Spaces in Duality 2.2- The Radon Transform for the Double Fibration: Principal Problems 2.3- Orbital Integrals 2.4- Examples of Radon Transforms for Homogeneous Spaces in Duality 3.- The Radon Transform on Two-Point Homogeneous Spaces 3.1- Spaces of Constant Curvature: Inversion and Support Theorems 3.2- Compact Two-Point Homogeneous Spaces: Applications 3.3- Noncompact Two-Point Homogeneous Spaces 3.4- Support Theorems Relative to Horocycles 4.- The X-Ray Transform on a Symmetric Space 4.1- Compact Symmetric Spaces: Injectivity and Local Inversion: Support Theorem 4.2- Noncompact Symmetric Spaces: Global Inversion and General Support Theorem 4.3- Maximal Tori and Minimal Spheres in Compact Symmetric Spaces 5.- Orbital Integrals 5.1- Isotropic Spaces 5.2- Orbital Integrals 5.3- Generalized Riesz Potentials 5.4- Determination of a Function from its Integral over Lorentzian Spheres 5.5- Orbital Integrands and Huygens' Principle 6.- The Mean-Value Operator 6.1- An Injectivity Result 6.2- Ásgeirsson's Mean-Value Theorem Generalized 6.3- John's Indentities 7.- Fourier Transforms and Distribution: A Rapid Course 7.1-The Topology of Spaces D(Rⁿ), E(Rⁿ) and S(Rⁿ) 7.2- Distribution 7.3- Convolutions 7.4- The Fourier Transform 7.5- Differential Operators with Constant Coefficients 7.6- Riesz Potentials 8.- Lie Transformation Groups and Differential Operators 8.1- Manifolds and Lie Groups 8.2- Lie Transformation Groups and Radon Transforms 9.- Bibliography 10.- Notational Conventions 11.- Index.

O autorze

Dr. Sigurdur Helgason has authored or co-authered 9 books and 94 papers. He has received numerous honors and awards, including the Steele Prize, awarded by the American Mathematical Society in 1988. He has been a professor at MIT since 1960.

Blurb

In this text, integral geometry deals with Radon’s problem of representing a function on a manifold in terms of its integrals over certain submanifolds—hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial differential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation. From the reviews: “Integral Geometry is a fascinating area, where numerous branches of mathematics meet together. The contents of the book is concentrated around the duality and double fibration, which is realized through the masterful treatment of a variety of examples. The book is written by an expert, who has made fundamental contributions to the area.” —Boris Rubin, Louisiana State University