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Let ''M'' be a real-analytic manifold and ''X'' its complexification.<!-- The definition of microfunctions here -->
Let ''M'' be a real-analytic manifold and ''X'' its complexification.<!-- The definition of microfunctions here -->


A microfunction can be used to define a hyper function. By definition, the sheaf of [[Sato's hyperfunction]]s on ''M'' is the restriction of the sheaf of microfunctions to ''M'', in parallel to the fact the sheaf of real-analytic functions on ''M'' is the restriction of the sheaf of holomorphic functions on ''X'' to ''M''.<ref>[[Masaki Kashiwara]] and [[Pierre Schapira]]: ''Sheaves on Manifolds.'' Springer-Verlag. Berlin Heidelberg New York.1990: {{ISBN|3-540-51861-4}}.</ref>
A microfunction can be used to define a hyper function. By definition, the sheaf of [[Sato's hyperfunction]]s on ''M'' is the restriction of the sheaf of microfunctions to ''M'', in parallel to the fact the sheaf of real-analytic functions on ''M'' is the restriction of the sheaf of holomorphic functions on ''X'' to ''M''.<ref>{{cite book |first1=Masaki |last1=Kashiwara |first2=Pierre |last2=Schapira |author-link2=Pierre Schapira |title=Sheaves on Manifolds |publisher=Springer-Verlag |location=Berlin |year=1990 |isbn=3-540-51861-4}}</ref>


==See also==
==See also==

Revision as of 14:01, 1 October 2019

Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunctions. As a research programme, it was started by Mikio Sato in 1959.[1]

Microfunction

Let M be a real-analytic manifold and X its complexification.

A microfunction can be used to define a hyper function. By definition, the sheaf of Sato's hyperfunctions on M is the restriction of the sheaf of microfunctions to M, in parallel to the fact the sheaf of real-analytic functions on M is the restriction of the sheaf of holomorphic functions on X to M.[2]

See also

References

  1. ^ Template:Cite article
  2. ^ Kashiwara, Masaki; Schapira, Pierre (1990). Sheaves on Manifolds. Berlin: Springer-Verlag. ISBN 3-540-51861-4.

Further reading