Synopsis
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.
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About the Author
Hans Duistermaat was a geometric analyst, who unexpectedly passed away in March 2010. His research encompassed many different areas in mathematics: ordinary differential equations, classical mechanics, discrete integrable systems, Fourier integral operators and their application to partial differential equations and spectral problems, singularities of mappings, harmonic analysis on semisimple Lie groups, symplectic differential geometry, and algebraic geometry. He was co-author of eleven books.
Duistermaat was affiliated to the Mathematical Institute of Utrecht University since 1974 as a Professor of Pure and Applied Mathematics. During the last five years he was honored with a special professorship at Utrecht University endowed by the Royal Netherlands Academy of Arts and Sciences. He was also a member of the Academy since 1982. He had 23 PhD students.
Review
"As observed in the introduction, this volume is essentially a TeX ed version of some lecture notes distributed by the Courant Institute of New York, corresponding to a course given by the author in the years 1970-71 about Fourier integral operators (FIO). Despite of many further results recently appeared, this book is still interesting, giving a quick and elegant introduction to the field, more adapted to non-specialists than, for example, the exposition in the monumental work of {L. Hoermander} [The analysis of linear partial differential operators, I--IV Springer-Verlag, Berlin. The contents are the following: after preliminaries on distributions, Fourier transform and wave front set in Chapter 1, the local theory of FIO is discussed in Chapter 2, having as particular case the calculus of pseudo-differential operators. Chapter 3 is devoted to symplectic differential geometry: symplectic vector spaces, symplectic manifolds, Lagrangian manifolds and links with classical mechanics and variational calculus. Chapter 4 concerns the global theory of FIO; pseudo-differential operators on manifolds are discussed as particular cases. Finally, Chapter 5 presents two applications: the Cauchy problem for strictly hyperbolic equations and caustics in oscillatory integrals. This last part of the book contains some interesting historical remarks, about connections with the work of Newton on optics and Huygens on wave fronts."
–Zentralblatt Math
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