Distributions

Theory and Applications

  • J.J. Duistermaat
  • J.A.C. Kolk

Part of the Cornerstones book series (COR)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. J. J. Duistermaat, J. A. C. Kolk
    Pages 1-15
  3. J. J. Duistermaat, J. A. C. Kolk
    Pages 17-32
  4. J. J. Duistermaat, J. A. C. Kolk
    Pages 33-44
  5. J. J. Duistermaat, J. A. C. Kolk
    Pages 45-49
  6. J. J. Duistermaat, J. A. C. Kolk
    Pages 51-58
  7. J. J. Duistermaat, J. A. C. Kolk
    Pages 59-63
  8. J. J. Duistermaat, J. A. C. Kolk
    Pages 65-70
  9. J. J. Duistermaat, J. A. C. Kolk
    Pages 71-82
  10. J. J. Duistermaat, J. A. C. Kolk
    Pages 83-90
  11. J. J. Duistermaat, J. A. C. Kolk
    Pages 91-113
  12. J. J. Duistermaat, J. A. C. Kolk
    Pages 115-136
  13. J. J. Duistermaat, J. A. C. Kolk
    Pages 137-152
  14. J. J. Duistermaat, J. A. C. Kolk
    Pages 153-176
  15. J. J. Duistermaat, J. A. C. Kolk
    Pages 177-220
  16. J. J. Duistermaat, J. A. C. Kolk
    Pages 221-236
  17. J. J. Duistermaat, J. A. C. Kolk
    Pages 237-270
  18. J. J. Duistermaat, J. A. C. Kolk
    Pages 271-285
  19. J. J. Duistermaat, J. A. C. Kolk
    Pages 287-310
  20. J. J. Duistermaat, J. A. C. Kolk
    Pages 311-320

About this book

Introduction

This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. Throughout the book, methods are developed to deal with formal calculations involving functions, series, and integrals that cannot be mathematically justified within the classical framework.

Key features:

• Many examples, exercises, hints, and solutions guide the reader throughout the text.

• Includes an introduction to distributions, differentiation, convergence, convolution, the Fourier transform, and spaces of distributions having special properties.

• Original proofs, which may be difficult to locate elsewhere, are given for many well-known results.

• The Fourier transform is transparently treated and applied to provide a new proof of the Kernel Theorem, which in turn is used to efficiently derive numerous important results.

• The systematic use of pullback and pushforward introduces concise notation.

• Emphasizes the role of symmetry in obtaining short arguments and investigates distributions that are invariant under the actions of various groups of transformations.

Distributions: Theory and Applications is aimed at advanced undergraduates and graduate students in mathematics, theoretical physics, and engineering, who will find this textbook a welcome introduction to the subject, requiring only a minimal mathematical background. The work may also serve as an excellent self-study guide for researchers who use distributions in various fields.

Keywords

Boundary value problem Distribution Fourier transform Sobolev space Sobolev spaces convolution dominated convergence theorem fractional integration and differentiation fundamental solution kernel theorem partial differential equation pullback and pushforward signal analysis tempered distribution test function

Authors and affiliations

  • J.J. Duistermaat
    • 1
  • J.A.C. Kolk
    • 2
  1. 1.Department of MathematicsUtrecht UniversityUtrechtNetherlands
  2. 2.Department of MathematicsUtrecht UniversityUtrechtNetherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-4675-2
  • Copyright Information Springer Science+Business Media, LLC 2010
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-4672-1
  • Online ISBN 978-0-8176-4675-2
  • About this book