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Fourier Integral Operators

  • Textbook
  • © 2011

Overview

Authors:
  1. J.J. Duistermaat

    1. Department of Mathematics, Utrecht University, Utrecht, Netherlands

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  • Based on author’s original lecture notes
  • Covers a broad range of topics
  • Useful for both practitioners and students
  • Applications put theory in context
  • Includes supplementary material: sn.pub/extras

Part of the book series: Modern Birkhäuser Classics (MBC)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Introduction

    • J. J. Duistermaat
    Pages 1-7

  3. Preliminaries

    • J. J. Duistermaat
    Pages 8-22

  4. Local Theory of Fourier Integrals

    • J. J. Duistermaat
    Pages 23-44

  5. Symplectic Differential Geometry

    • J. J. Duistermaat
    Pages 45-90

  6. Global Theory of Fourier Integral Operators

    • J. J. Duistermaat
    Pages 91-112

  7. Applications

    • J. J. Duistermaat
    Pages 113-137

  8. Back Matter

    Pages 138-142
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Keywords

About this book

More than twenty years ago I gave a course on Fourier Integral Op­ erators at the Catholic University of Nijmegen (1970-71) from which a set of lecture notes were written up; the Courant Institute of Mathematical Sciences in New York distributed these notes for many years, but they be­ came increasingly difficult to obtain. The current text is essentially a nicely TeXed version of those notes with some minor additions (e.g., figures) and corrections. Apparently an attractive aspect of our approach to Fourier Integral Operators was its introduction to symplectic differential geometry, the basic facts of which are needed for making the step from the local definitions to the global calculus. A first example of the latter is the definition of the wave front set of a distribution in terms of testing with oscillatory functions. This is obviously coordinate-invariant and automatically realizes the wave front set as a subset of the cotangent bundle, the symplectic manifold in which the global calculus takes place.

Reviews

From the reviews:

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.

“Duistermaat’s Fourier Integral Operators had its genesis in a course the author taught at Nijmegen in 1970. … For the properly prepared and properly disposed mathematical audience Fourier Integral Operators is a must. … it is a very important book on a subject that is both deep and broad.” (Michael Berg, The Mathematical Association of America, May, 2011)

Authors and Affiliations

  • Department of Mathematics, Utrecht University, Utrecht, Netherlands

    J.J. Duistermaat

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.

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