
Author(s)  Imai, Isae 

Title  Applied Hyperfunction Theory 
Publisher  Kluwer Academic Publishers 
Year of publication  1992 
Reviewed by  Vasile Postolica 
This book which deals with the hyperfunction theory applicable to problems of engineering and science is the synthesis of the author's studies undertaken from August 1975 through the agency of the series entitled "Invitation to Fluid Mathematics II Hyperfunction Theory" in the Journal Mathematical Sciences of Sciencesha Publishing Company.
In Chapter 1 Sato's hyperfunction is considered heuristically as a vortex layer familiar in hydrodynamics and the construction of a hyperfunction theory based on this interpretation is discussed here. Chapter 2 is devoted to addition, subtraction, differentiation, and integration of hyperfunctions. In Chapter 3 one introduces the concepts of even, odd, real, and imaginary hyperfunctions and the formal product. Chapter 4 deals with the parameterdependent hyperfunctions. In Chapter 5 a general theory of Fourier transformation of hyperfunctions is developed and in Chapter 6 we find the Fourier transforms of powertype hyperfunctions. Chapter 7 is devoted to upper, lower, left, and right hyperfunctions and in Chapter 8 one gives criteria for the existence of Fourier transforms and for their regularity. In Chapter 9 the asymptotic behaviours for some Fourier transforms are studied.
The concept of a standard generating function is introduced in Chapter 10 in which it is shown that the theory of Fourier series is in fact absorbed in that of Fourier transforms. The theorems on analytic continuation and the identity for hyperfunctions are presented in Chapter 11 where we find a reinterpretation of ordinary functions as hyperfunctions. In Chapter 12 is given an appropriate definition of product when the hyperfunctions do not share singular points and both are upper (lower) hyperfunctions. Convolutions for hyperfunctions are discussed in Chapter 13 and the special cases of convolutions for periodic hyperfunctions are considered in Chapter 14. Chapter 15 deals with the Hilbert transformation which is important for applications and in Chapter 16 PoissonSchwarz integral formulae are given in order to be applied in the solution of certain types of integral equations. Integral equations of convolution type solved through the agency of hyperfunctions are given in Chapter 17. The last chapter shows that the theory of Laplace transformation is absorbed in that of Fourier transformations. The "Epilogue" contains important comments.
It is the merit of the author that the book can be perused without recourse to references. For this reason, we find only 17 bibliographical references, although the content is richly presented. In addition, 11 appendices facilitate the reading together with the corresponding index. In our opinion, this volume is one of the best in the above field.