Distributions

This text offers an application-oriented introduction to the theory of distributions. It presents distributions as a natural method of analysis from a mathematical and physical point of view. The subject is motivated by many exercises, hints and solutions.

This text offers a uniquely application-oriented introduction to the theory of distributions. It presents distributions as a natural method of analysis from both a mathematical and physical point of view. The discussion emphasizes applications to the general study of differential equations and in particular, linear partial differential equations. In this book the subject is motivated by many exercises, hints and solutions that guide the student along a path requiring only a minimal mathematical background. Advanced undergraduate and graduate students in mathematics, theoretical physics and engineering will find this textbook a welcome introduction to the theory of distributions, a subject often requiring a strong mathematical background.

Emphasizes applications to physical phenomena such as optics, quantum mechanics, quantum field theory and partial differential equations Includes many exercises with detailed hints and solutions Approaches the subject from a unique, application-oriented point of view Includes supplementary material: sn.pub/extras

Auteur

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/

Contenu
Motivation.- Test Functions.- Distributions.- Differentiation of Distributions.- Convergence of Distributions.- Taylor Expansion in Several Variables.- Localization.- Distributions with Compact Support.- Multiplication by Functions.- Transposition: Pullback and Pushforward.- Convolution of Distributions.- Fundamental Solutions.- Fractional Integration and Differentiation.- Fourier Transform.- Distribution Kernels.- Fourier Series.- Fundamental Solutions and Fourier Transform.- Supports and Fourier Transform.- Sobolev Spaces.- Appendix: Integration.- Solutions to Selected Problems.