Demidov A.S. Generalized Functions in Mathematical Physics: Main Ideas and Concepts

Nova Science Publishers, 2013. — 153 p. — (Horizons in World Physics). — ISBN 1560729058.In a nutshell the book gives an interconnected presentation of a some basic ideas, concepts, results of the theory of generalized functions (first of all, in the framework of the theory of distributions) and equations of mathematical physics. Chapter 1 acquaints the reader with some initial elements of the language of distributions in the context of the classical equations of mathematical physics (the Laplace equation, the heat equation, the string equation). Here some basic facts from the theory of the Lebesgue integral are presented, the Riesz spaces of integrable functions are introduced. In the section devoted to the heat equation, the student of mathematics can get familiar with the method of dimensionality and similarity, which is not usually included in the university program for mathematicians, but which is rather useful on the initial stage of study of the problems of mathematical physics. Chapter 2 is devoted to the fundamentals of the theory of distributions due to L. Schwartz. Section 16 is the most important. The approach to some topics can also be interesting for the experts. Chapter 3 acquaints the reader with some modern tools and methods for the study of linear equations of mathematical physics. The basics of the theory of Sobolev spaces, the theory of pseudodifferential operators, the theory of elliptic problems (including some elementary results concerning the index of elliptic operators) as well as some other problems connected in some way with the Fourier transform (ordinary functions and distributions) are given here.
Now I would like to say a few words concerning the style of the book. A part of the material is given according to the scheme: definition — theorem — proof. This scheme is convenient for presenting results in clear and concentrated form. However, it seems reasonable to give a student the possibility not only to study a priori given definitions and proofs of theorems, but also to discover them while considering the problems involved. A series of sections serves this purpose. Moreover, a part of the material is given as exercises and problems. Thus, reading the book requires, in places, a certain effort. However, the more difficult problems are supplied with hints or references. Problems are marked by the letter P (hint on Parking for the solution of small Problems). The importance of numerous notes is essentially connected with a playful remark by V.F. Dyachenko: “The most important facts should be written in notes, since only those are read”. The notes are typeset in a small font and located in the text immediately after the current paragraph.True PDFA5 format