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| '''Algebraic analysis''' is an area of ] that deals with systems of linear ]s by using ] and ] to study properties and generalizations of functions such as ]s and microfunctions. As a research programme, it was started by ] in 1959.<ref>{{cite article|title=Professor Mikio Sato and Microlocal Analysis|author1=Masaki Kashiwara|author2=Takahiro Kawai|journal=PRIMS|volume=47|issue=1|year=2011|url=http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=47&iss=1&rank=2|doi=10.2977/PRIMS/29|via=EMS-PH}}</ref> | '''Algebraic analysis''' is an area of ] that deals with systems of linear ]s by using ] and ] to study properties and generalizations of functions such as ]s and microfunctions. As a research programme, it was started by ] in 1959.<ref>{{cite article|title=Professor Mikio Sato and Microlocal Analysis|author1=Masaki Kashiwara|author2=Takahiro Kawai|journal=PRIMS|volume=47|issue=1|year=2011|url=http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=47&iss=1&rank=2|doi=10.2977/PRIMS/29|via=EMS-PH}}</ref> | ||
| == |
== Microfunction == | ||
| {{expand section|date=September 2019}} | |||
| {{Reflist}} | |||
| 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 ]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''. | |||
| ==See also== | ==See also== | ||
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| *] | *] | ||
| *] | *] | ||
| == References == | |||
| *] and ]: ''Sheaves on Manifolds.'' Springer-Verlag. Berlin Heidelberg New York.1990: {{ISBN|3-540-51861-4}}. | |||
| ==Further reading== | ==Further reading== | ||
Revision as of 05:04, 1 October 2019
Not to be confused with the common phrase "algebraic analysis of ", meaning "the algebraic study of "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.
Microfunction
 | This section needs expansion. You can help by adding to it. (September 2019) |
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.
See also
- Hyperfunction
- D-module
- Microlocal analysis
- Generalized function
- Edge-of-the-wedge theorem
- FBI transform
- Localization of a ring
- Vanishing cycle
- Gauss–Manin connection
- Differential algebra
- Perverse sheaf
- Mikio Sato
- Masaki Kashiwara
- Lars Hörmander
References
- Masaki Kashiwara and Pierre Schapira: Sheaves on Manifolds. Springer-Verlag. Berlin Heidelberg New York.1990: ISBN 3-540-51861-4.
Further reading
 | This mathematical analysis–related article is a stub. You can help Misplaced Pages by expanding it. |