Format:
Online-Ressource (XI, 142p. 9 illus, digital)
ISBN:
9780817681081
Series Statement:
Modern Birkhäuser Classics
Content:
Preface -- 0. Introduction -- 1. Preliminaries -- 1.1 Distribution densities on manifolds -- 1.2 The method of stationary phase -- 1.3 The wave front set of a distribution -- 2. Local Theory of Fourier Integrals -- 2.1 Symbols -- 2.2 Distributions defined by oscillatory integrals -- 2.3 Oscillatory integrals with nondegenerate phase functions -- 2.4 Fourier integral operators (local theory) -- 2.5 Pseudodifferential operators in Rn -- 3. Symplectic Differential Geometry -- 3.1 Vector fields -- 3.2 Differential forms -- 3.3 The canonical 1- and 2-form T* (X) -- 3.4 Symplectic vector spaces -- 3.5 Symplectic differential geometry -- 3.6 Lagrangian manifolds -- 3.7 Conic Lagrangian manifolds -- 3.8 Classical mechanics and variational calculus -- 4. Global Theory of Fourier Integral Operators -- 4.1 Invariant definition of the principal symbol -- 4.2 Global theory of Fourier integral operators -- 4.3 Products with vanishing principal symbol -- 4.4 L2-continuity -- 5. Applications -- 5.1 The Cauchy problem for strictly hyperbolic differential operators with C-infinity coefficients -- 5.2 Oscillatory asymptotic solutions. Caustics -- References.
Content:
This volume is a useful introduction to the subject of Fourier integral operators and is based on the author's classic set of notes. Covering a range of topics from Hörmander’s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes applications to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics. This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, resp. WKB-methods. Familiarity with analysis (distributions and Fourier transformation) and differential geometry is useful. Additionally, this book is designed for a one-semester introductory course on Fourier integral operators aimed at a broad audience. This book remains a superb introduction to the theory of Fourier integral operators. While there are further advances discussed in other sources, this book can still be recommended as perhaps the very best place to start in the study of this subject. —SIAM Review This book is still interesting, giving a quick and elegant introduction to the field, more adapted to nonspecialists. —Zentralblatt MATH The book is completed with applications to the Cauchy problem for strictly hyperbolic equations and caustics in oscillatory integrals. The reader should have some background knowledge in analysis (distributions and Fourier transformations) and differential geometry. —Acta Sci. Math.
Note:
"Reprint of the 1996 edition
,
Includes bibliographical references (p. 138-142)
,
""Fourier Integral Operators""; ""Contents""; ""Preface""; ""Fourier Integral Operators""; ""Chapter 0: Introduction ""; ""Chapter 1: Preliminaries ""; ""1.1. Distribution densities on manifolds""; ""1.2. The method of stationary phase""; ""1.3. The wave front set of a distribution""; ""Chapter 2: Local Theory of Fourier Integrals ""; ""2.1. Symbols""; ""2.2. Distributions denned by oscillatory integrals""; ""2.3. Oscillatory integrals with nondegenerate phase functions""; ""2.4. Fourier integral operators (local theory)""; ""2.5. Pseudodifferential operators in Râ?¿""
,
""Chapter 3: Symplectic Differential Geometry """"3.1. Vector fields""; ""3.2. Differential forms""; ""3.3. The canonical 1- and 2-form in T*(X)""; ""3.4. Symplectic vector spaces""; ""3.5. Symplectic differential geometry""; ""3.6. Lagrangian manifolds""; ""3.7. Conic Lagrangian manifolds""; ""3.8. Classical mechanics and variational calculus""; ""Chapter 4: Global Theory of Fourier Integral Operators ""; ""4.1. Invariant definition of the principal symbol""; ""4.2. Global theory of Fourier integral operators""; ""4.3. Products with vanishing principal symbol""; ""4.4. L²-continuity""
,
""Chapter 5: Applications """"5.2. Oscillatory asymptotic solutions. Caustics""; ""5.1. The Cauchy problem for strictly hyperbolic differential operators with Câ?ž coefficients""; ""References""
Additional Edition:
ISBN 9780817681074
Additional Edition:
Buchausg. u.d.T. Duistermaat, Johannes J., 1942 - 2010 Fourier integral operators Boston, Mass. [u.a.] : Birkhäuser, 2011 ISBN 9780817681074
Additional Edition:
Erscheint auch als Druck-Ausgabe Duistermaat, Johannes J., 1942 - 2010 Fourier integral operators Boston [u.a.] : Birkhäuser, 1996 ISBN 0817638210
Additional Edition:
ISBN 3764338210
Language:
English
Subjects:
Mathematics
Keywords:
Fourier-Integraloperator
;
Fourier-Integraloperator
DOI:
10.1007/978-0-8176-8108-1
URL:
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URL:
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