Introduction to Hyperfunctions and Their Integral Transforms - by Urs Graf (Hardcover)

About the Book

This textbook presents an introduction to the subject of generalized functions and their integral transforms by an approach based on the theory of functions of one complex variable. It includes many concrete examples.

Book Synopsis

This textbook is about generalized functions and some of their integral tra- forms in one variable. It is based on the approach introduced by the Japanese mathematician Mikio Sato. We mention this because the term hyperfunction that Sato has given to his generalization of the concept of function is sometimes used today to denote generalized functions based on other approaches (distributions, Mikusinski s operators etc. ). I have written this book because I am delighted by the intuitive idea behind Sato s approach which uses the classical complex fu- tion theory to generalize the notion of function of a real variable. In my opinion, Sato swayofintroducingthegeneralizedconceptofafunctionislessabstractthan the one of Laurent Schwartz who de?nes his distributions as linear functionals on some space of test functions. On the other hand, I was quickly led to recognize that very few colleagues (mathematicians included) knew anything about Satos s approach. PerhapsSato andhis schoolis not entirelyblamelessfor this state of - fairs. For severaldecades no elementary textbook addressinga wider audience was available (at least in English). Zealots delighted by the appealing intuitive idea of the approach have probably found their enthusiasm rapidly diminished because of the adopted style of exposition and the highly ambitious abstract mathematical concepts used in the available books and articles. Fortunately, some years ago, I found Isac Imai s Book Applied Hyperfunction Theory which explains and applies Sato s hyperfunctions in a concrete, but nontrivial way, and thereby reveals their computational power."

From the Back Cover

This textbook presents an elementary introduction to generalized functions by using Sato's approach of hyperfunctions which is based on complex function theory. This very intuitive and appealing approach has particularly great computational power.

The concept of hyperfunctions and their analytic properties is introduced and discussed in detail in the first two chapters of the book. Thereafter the focus lies on generalizing the (classical) Laplace, Fourier, Hilbert, Mellin, and Hankel transformations to hyperfunctions. Applications to integral and differential equations and a rich variety of concrete examples accompany the text throughout the book.

Requiring only standard knowledge of the theory of complex variables, the material is easily accessible for advanced undergraduate or graduate students. It serves as well as a reference for researchers in pure and applied mathematics, engineering and physics.

Review Quotes

From the reviews:

"The intended audience of the book is advanced undergraduate students, graduate students, and research professionals in mathematical sciences. The book can serve as a research reference or a supplement to courses or seminars. ... this book provides an excellent presentation of integral transforms of hyperfunctions with examples of applications to integral equations and ordinary and partial differential equations. ... would be successful as a senior undergraduate- or graduate-level text in mathematics, physics, and engineering, as well as a research reference for professionals in those fields." (Lokenath Debnath, SIAM Review, Vol. 53 (1), 2011)

"This is an introductory and elementary textbook about Sato's hyperfunctions in one variable and some of their integral transforms. ... mainly addressed to applied mathematicians, physicists and engineers, it may also be useful for pure mathematicians seeking a first encounter with Sato's beautiful conception of generalized functions. ... The only prerequisites for the reader are elementary notions from complex function theory of one variable and some familiarity with the classical Laplace transform. This makes the material of the book accessible even to undergraduate students." (Jasson Vindas, Mathematical Reviews, Issue 2012 a)

"This monograph aims at introducing the theory of hyperfunctions and some of their integral transforms to a wide range of readers, since the author recognized that the noble idea of hyperfunction was known to only few mathematicians." (Dohan Kim, Zentralblatt MATH, Vol. 1201, 2011)