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Zbl 1201.46001
Graf, Urs
Introduction to hyperfunctions and their integral transforms. An applied and computational approach.
(English)
[B] Basel: Birkhäuser. xi, 415~p. EUR~59.95/net; \sterling~53.99; SFR~99.00 (2010). ISBN 978-3-0346-0407-9/hbk

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. He tried to make the prerequisites for this book minimal, relying only on basic notions of complex function theory and of the classical Laplace and Fourier transformations. Hyperfunctions were introduced by the Japanese mathematician Mikio Sato in 1958 as the difference between (the boundary values of) two holomorphic (defining) functions, $F^+$ in the upper half plane and $F^-$ in the lower half plane [cf.\ {\it M.\,Sato}, J.~Fac.\ Sci.\ Univ.\ Tokyo Sect.\ I 8, 139--193 (1959; Zbl 0087.31402); ibid.\ 387--436 (1960; Zbl 0097.31404)]. In his famous book [The road to reality.\ A complete guide to the laws of the universe" (New York:\ Alfred A.\ Knopf, Inc.) (2005; Zbl 1188.00007)], {\it R.\,Penrose} mentioned their importance: In trying to generalize the notion of function' as far as we can away from the apparently very restrictive notion of an analytic' or holomorphic' function -- the type of function that would have made Euler happy -- we have come to the extremely general and flexible notion of {\it hyperfunction}. But hyperfunctions are themselves defined, in a basically very simple way, in terms of these very same Eulerian' holomorphic functions that we thought we had reluctantly abandoned. In my view, this is one of the supreme magical achievements of complex numbers. If only Euler had been alive to appreciate this wondrous fact! After a brief introduction to hyperfunctions of one variable in Chapters I and II, following {\it I.\,Imai}'s [Applied Hyperfunction Theory'' (Math.\ Appl., Jap.\ Ser.\ 8; Dordrecht:\ Kluwer) (1992; Zbl 0787.46033 )], the author deals with their integral transforms, including Laplace, Fourier, Hilbert, Mellin and finally Hankel transforms, emphasizing the computational aspect and application. To make this book accessible to a larger audience, he skipped some important topics, as he mentions in the preface. No sheaves and other `microlocal' concepts are mentioned and the intended message is rather conveyed through many concrete examples. The treatment of hyperfunctions of several variables is omitted as well.
[Dohan Kim (Seoul)]
MSC 2000:
*46-02 Research monographs (functional analysis)
44-02 Research monographs (integral transforms)
46F15 Hyperfunctions, analytic functionals
46F12 Integral transforms in distribution spaces

Keywords: hyperfunction; Laplace transform; Fourier transform; Hilbert transform; Mellin transform; Hankel transform; monograph

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