zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An introduction to Sato’s hyperfunctions. Transl. from the Japanese by Mitsuo Morimoto. (English) Zbl 0811.46034
Translations of Mathematical Monographs. 129. Providence, RI: American Mathematical Society (AMS). vii, 273 p. $ 87.00 (1993).
The book is a translation of the author’s book on hyperfunctions which appeared in Japanese in 1976. As the title indicates it gives an introduction to the theory of hyperfunctions. The first two chapters recall some facts about complex function theory and analytic functionals, especially the Köthe duality theorem. Hyperfunctions of one variable on a locally closed subset A of are defined in the next chapter as elements of H A 1 (W,𝒪)=𝒪(WA)/𝒪(W). The usual operations of analysis are defined and the hyperfunctions with compact support are identified with analytic functionals via the duality theorem. The necessary tools from sheaf theory and cohomology theory with coefficients in 𝒪 are treated in the next two chapters and in chapter six the Martineau duality theorem is proved. This gives in the n-dimensional case the identification of hyperfunctions with compact support with analytic functionals. The space of hyperfunctions in n dimensions is defined, as in Satos original paper, as relative cohomology group with coefficients in 𝒪 and it is proved that the sheaf of hyperfunctions is flabby. The representation as classes of holomorphic functions is given by Čech-cohomology and the operator of taking boundary values for holomorphic functions is defined. The last two chapters give an introduction to the theory of microfunctions and a proof of Satos regularity theorem and of the edge- of-the-wedge theorem are given as applications. These two chapters can serve as an introduction to microlocal analysis and motivate the more complicated treatment in more recent books by e.g. M. Kashiwara and P. Schapira, ‘Sheaves on manifolds’, Berlin (1990; Zbl 0709.18001). Many concrete examples illustrate the theory and make the book a valuable text for beginners.

46F15Hyperfunctions, analytic functionals
46F20Distributions and ultradistributions as boundary values of analytic functions
46M20Methods of algebraic topology in functional analysis
46-01Textbooks (functional analysis)
32A45Hyperfunctions (complex analysis)