Distributions: Theory and Applications (Cornerstones) - Hardcover

Duistermaat, J.J.; Kolk, Johan A.C.

Hardcover

ISBN 10: 0817646728 ISBN 13: 9780817646721

Publisher: Birkhäuser, 2010

Synopsis

I am sure that something must be found. There must exist a notion of generalized functions which are to functions what the real numbers are to the rationals (G. Peano, 1912) Not that much effort is needed, for it is such a smooth and simple theory (F. Tre`ves, 1975) In undergraduate physics a lecturer will be tempted to say on certain occasions: “Let ?. x/ be a function on the line that equals 0 away from 0 and is in?nite at 0 in such a way that its total integral is 1. The most important property of ?. x/ is exempli?ed Z by the identity 1 . x/?. x/ dx D . 0/; 1 where is any continuous function of x. ” Such a function ?. x/ is an object that one frequently would like to use, but of course there is no such function, because a function that is 0 everywhere except at one point has integral 0. All the same, it is important to realize what our lecturer is trying to accomplish: to describe an object in terms of the way it behaves when integrated against a function. It is for such purposes that the theory of distributions, or “generalized functions,” was created. It can be formulated in all dimensions, its mathematical scope is vast, and it has revolutionized modern analysis.

About this title

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 with 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.

Johan Kolk has published in the areas of harmonic analysis on semisimple Lie groups, the theory of distributions, and classical analysis. Jointly with Duistermaat, he has written four books: besides the present one, one on Lie groups, and another on multidimensional real analysis. Until his retirement in 2009, he was affiliated with the Mathematical Institute of Utrecht University. For more information, see http://www.staff.science.uu.nl/~kolk0101/

From the Back Cover

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.

Edition details

• Publisher: Birkhäuser
• Publication date: 2010
• Language: English
• ISBN 10: 0817646728
• ISBN 13: 9780817646721
• Binding: Hardcover
• Number of pages: 461