Hyperfunction (math)
In mathematics , a hyperfunction is a generalization of functions as a jump from one holomorphic function to another holomorphic function on a given limit :
history
There are different approaches to the theory of hyperfunctions. In 1958, Mikio Satō was the first to introduce hyperfunctions , primarily based on the work of Alexander Grothendieck . He defined them in an abstract sense as boundary values on the real axis. So by hyperfunctions Sato understood pairs of functions that are analytic for or for modulo the pair , where is a whole analytic function . In a second work he expanded the concept of hyperfunctions to functions in the with the help of the sheaf cohomology theory . Sato's access to hyperfunctions in the is quite cumbersome. With the help of the theory of analytical functionals , André Martineau developed another approach to hyperfunctions.
Analytical functional
Let be a compact subset . In the following, the space of the functions is referred to, which are analytically thus whole functions . The topological dual space is the space of the applied analytical functionals. That is, it is the space of linear forms on which all environments of the inequality
meet for everyone . The space of the applied analytical functionals is thus a distribution space . The topological vector space of the smooth functions is denoted by. Since it is a dense subspace, the distribution space can be identified with a subspace of .
definition
After Mikio Sato
According to Sato, a hyperfunction in one dimension is represented by a pair of holomorphic functions that are separated by an edge . In most cases it is part of the real number axis. In this case, the lower complex half- plane is defined in an open subset and the upper complex half-plane is defined in an open subset. A hyperfunction is the "jump" from to over the edge .
After André Martineau
Be an open and bounded subset. Then the space of Hyper functions on by
Are defined.
Examples
literature
- Lars Hörmander: The Analysis of Linear Partial Differential Operators I , Springer-Verlag, Second Edition, ISBN 3-540-52345-6 , Chapter IX
- A. Kaneko: Hyperfunction . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Eric W. Weisstein : Hyperfunction . In: MathWorld (English).