Operational calculus. A theory of hyperfunctions. Transl. from the Japanese
In the mathematical literature there are many books in operational calculus. All they are very similar. We shall compare this one only with he book of \textit{J. Mikusiński} [Operational calculus (1957; Zbl 0078.113), (1959; (Zbl 0088.330) and the second english edition of Volume I (1983; Zbl 0532.44003)] as the basic literature in this domain. The author in the first five chapters, in distinction from J. Mikusiński, constructs and develops a subring \(C_ H\) of the field M of Mikusinski operators which consists of convolution quotients \(f/\ell^ n\) where \(f\in C_{[0,\infty)}\) and \(\ell\) is the integral operator. This ring is sufficient for solving linear ordinary differential equations with constant coefficients. Only in the sixth chapter he introduces the Titchmarsh convolution theorem and the field M of Mikusiński operators. His proof of the Titchmarsh's theorem is based on Liouville's theorem in analytic function theory and is a little shorter than the one by J. Mikusiński. The proof by J. Mikusiński is based exclusively on the methods of functions of a real variable and uses the Phragmén's theorem. From the elements of M, the author analyses separately shift and diffusion operators. Applications to partial differential equations are deduced to the wave, telegraph and heat equation. The book is written correctly, clearly and in an economical way; it has only 170 pages.
scientific article; zbMATH DE number 3861769