Book review

This book gives an intelligible exposition of generalized functions theory based on Sato's hyperfunctions, which are essentially the 'jumps of boundary values of analytic functions under passage cross real axes. More precisely, Sato's generalized functions (hyperfunctions) are equivalence classes of analytical on the set <formula> functions differing from one another by entire functions; an usual 'normal' function considered as a difference of limit upper and lower values of analytical on the set <formula> function defined with some integral of Cauchy type turns out to be Sato's hyperfunction; analogous assertion holds for Schwarz distributions. Sato's hyperfunctions are now widely known and applied both in different branches of mathematics and mathematical physics and in engineering and technology. There is a row books devoted to Sato's hyperfunctions, for example, books written by P. Schapira or K. Yosida, however these books are destined for mathematicians and hardly can be used for initial acquaintance with this theory. And so this new book devoted to Sato's hyperfunctions fills an important gap in mathematical literature - it is an elementary introduction and throws light on basic moments in the theory.

The author of the book, Isao Imai is a specialist in hydrodynamics and so he gives the main part of this book as a researcher who applies theory of hyperfunctions in his field. This fact makes this book to be accessible both mathematicians and different users of mathematics including undergraduates. However this fact makes this book to be more interesting for mathematicians because the author in many important points of theory gives some unexpected and informal comments that adorn this book and make it to be nonstandard.

The book contains a small Preface and Epilogue in which the author presents the history of creation of book, 18 chapters, References and 11 appendices. Chapter 1 'Introduction' is devoted to basic definitions in the hyperfunction theory and formal descriptions of relations between usual functions, hyperfunctions and other types of generalized functions; in this chapter one can find a curious hydrodynamics interpretation of hyperfunctions too. Chapter 2 'Operations on hyperfunctions' describes linear algebraic operations with hyperfunctions, their differentiation and integration; of course, in this chapter the product of hyperfunctions is defined only in the case when one of multipliers is an entire function. Chapter 3 'Basic hyperfunctions' is devoted to the description of classes of even and odd, real and imaginary functions as well as hyperfunctions 'generating' with function of Heaviside, power functions, logarithmic function and their products; besides the problem of division of hyperfunctions are discussed. Chapter 4 'Hyperfunctions depending on parameters' deals with convergence in the space of hyperfunctions and continuity of the main operations with hyperfunctions including differentiation and integration; a major part of chapter deals with calculations with concrete hyperfunctions. The following two chapter 'Fourier transformation' and 'Fourier transformation of power-type hyperfunctions' presents the definition of Fourier transformation for hyperfunctions its main properties, generalizations of Fourier formula, example of calculation and Pourier transforms of concrete functions.

Chapter 7 'Upper (lower)-type hyperfunctions' is devoted to two important special classes of hyperfunctions that have 'representators' vanishing in either upper or lower half-plane; the properties of these functions and numerous their examples are discussed in details. Chapter 8 'Fourier transforms - existence and regularity' gives some sufficient conditions for the existence of Fourier transforms for hyperfunctions of special (exponential) type, playing an important role in the theory. Chapter 9 'Fourier transforms - asymptotic behaviour' deals with the behaviour of Fourier transforms at the infinity; in particular, the Riemann-Lebesgue theorem for hyperfunction is considered.

Chapter 10 'Periodic hyperfunctions and Fourier series' presents elements of theory of periodic hyperfunctions and Fourier series the analysis of which is reduced to Fourier transformation; the problem of calculation of Fourier series for different concrete functions its considered in details. Chapter 11 'Analytic continuation and projection of hyperfunctions' deals with some elementary decompositions of the space of hyperfunctions into 'direct sums' subspaces; these decompositions allow to divide 'singularities' of hyperfunctions from one another and play an important role in many problems for hyperfunctions. Chapter 12 'Product of hyperfunctions' is devoted to general discussion of possibilities to introduce the product of two or several hyperfunctions; as a result the product of two hyperfunctions is defined if these functions have no common singular points or if they are upper-type or lower-type ones.

Chapter 13 'Convolutions of hyperfunctions' deals with convolution of hyperfunctions of different types, sufficient conditions for the existence, basic properties, including relations with Fourier transformation. and Parseval's theorem analog. Chapter 14 'Convolutions of periodic hyperfunctions' continues theory of convolutions but for periodic hyperfunctions and their Fourier series. Chapters 15 'Hilbert transforms and conjective hyperfunctions' and 16 'Poisson-Schwarz integral formulae' are devoted to generalizations of basic facts on Hilbert transformation and boundary value problems for Laplace equation or for analytical functions; as application some integral equations with kernels of Cauchy-type in classes of hyperfunctions are considered. Chapter 17 'Integral equations' is devoted to applications theory of hyperfunctions to convolution integral equations, Volterra integral equations, Abel integral equations and so on. The last chapter 18 'Laplace transforms' contains a short sketch theory of Laplace transformation for hyperfunctions.

References is very short and contains only 17 items. Appendices gives tables of definitions for different functions and corresponding hyperfunctions studied in different parts of the book, their Hilbert transforms Fourier, Laplace, Cosine and sine transforms, their convolutions and so on.

As may be seen the book contains reach information on abstract theory of hyperfunctions and its applications to different problems of analysis. One can use this book as a textbook in field and well as a reference book of elementary hyperfunctions.