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. The book is motivated by many exercises, hints, and solutions that guide the reader along a path requiring only a minimal mathematical background.
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/
Preface.- Standard Notation.- 1 Motivation .- Problems.- 2 Test Functions.- Problems.- 3 Distributions.- Problems.- 4 Differentiation of Distributions.- Problems.- 5 Convergence of Distributions.- Problems.- 6 Taylor Expansion in Several Variables.- Problems.- 7 Localization.- Problems.- 8 Distributions with Compact Support.- Problems.- 9 Multiplication by Functions.- Problems.- 10 Transposition: Pullback and Pushforward.- Problems.- 11 Convolution of Distributions.- Problems.- 12 Fundamental Solutions.- Problems.- 13 Fractional Integration and Differentiation .- 13.1 The Case of Dimension One.- 13.2 Wave Family.- 13.3 Appendix: Euler's Gamma Function.- Problems.- 14 Fourier Transform.- Problems.- 15 Distribution Kernels.- Problems.- 16 Fourier Series.- Problems.- 17 Fundamental Solutions and Fourier Transform.- 17.1 Appendix: Fundamental Solution of .I?/k.- Problems.- 18 Supports and Fourier Transform.- Problems.- 19 Sobolev Spaces.- Problems.- 20 Appendix: Integration.- 21 Solutions to Selected Problems.- References.- Index of Notation.- Index.