The theory of distributions is most often presented as L. Schwartz originally presented it: as a theory of the duality of topological vector spaces. Although this is a sound approach, it can be difficult, demanding deep prior knowledge of functional analysis. The more elementary treatments that are available often consider distributions as limits of sequences of functions, but these usually present the theoretical foundations in a form too simplified for practical applications.
Distributions, Integral Transforms and Applications offers an approachable introduction to the theory of distributions and integral transforms that uses Schwartz's description of distributions as linear continous forms on topological vector spaces. The authors use the theory of the Lebesgue integral as a fundamental tool in the proofs of many theorems and develop the theory from its beginnings to the point of proving many of the deep, important theorems, such as the Schwartz kernel theorem and the Malgrange-Ehrenpreis theorem. They clearly demonstrate how the theory of distributions can be used in cases such as Fourier analysis, when the methods of classical analysis are insufficient.
Accessible to anyone who has completed a course in advanced calculus, this treatment emphasizes the remarkable connections between distributional theory, classical analysis, and the theory of differential equations and leads directly to applications in various branches of mathematics.
Definitions and Preliminaries
Local Properties of Distribution
Tensor Products and Convolution Products
Particular Types of Distribution and Cauchy Transforms
Tempered Distributions and Fourier Transforms
Orthogonal Expansions of Distribution
Appendix: Sequential Completeness of some Spaces
Notes and References to the Literature
Index of Symbols
"This interesting book is a handy self-contained introduction to the theory of distributions and its main applications. I recommend this book to PhD students and researchers as a fairly complete introductory course to distribution theory and integral transforms."
-Mathematical Reviews, Issue 2004f