Operational calculus. A theory of hyperfunctions. Transl. from the Japanese
type: Item
instance of: scholarly article
title: Operational calculus. A theory of hyperfunctions. Transl. from the Japanese
zbMATH Open document ID: 0542.44001
published in: Applied Mathematical Sciences
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 44A40
Mathematics Subject Classification ID: 44-02
Mathematics Subject Classification ID: 34A25
zbMATH DE Number: 3861769
zbMATH Keywords: subring
zbMATH Keywords: Mikusinski operators
zbMATH Keywords: convolution quotients
zbMATH Keywords: Titchmarsh convolution theorem
zbMATH Keywords: Phragmén's theorem
zbMATH Keywords: diffusion operators
author: Kosaku Yosida
reviewed by: B. Stanković
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3861769