Hyperfunction

In mathematics, a hyper- function is a generalization of functions as a jump of a holomorphic function on another holomorphic function on a given limit:

History

There are different approaches to the theory of hyperfunctions. Mikio Sato introduced in 1958 as the first mainly based on the work of Alexander Grothendieck hyperfunctions. He defined it in an abstract sense as boundary values ​​on the real axis. So Sato hyperfunctions understood by pairs of functions which are, respectively, for modulo the pair, with a very analytic function are analytically. In a second work, he advanced with the help of Garbenkohomologietheorie the concept of hyperfunctions on functions. This approach of Sato for Hyper functions is quite cumbersome. So André Martineau developed using the theory of analytic functionals further access to the hyperfunctions.

Analytical functional

Let be a compact subset. Hereinafter will be referred to with the space of the functions, which are thus to all analytical functions. The topological dual space is the space of analytic functionals carried on. That is, it is the space of linear forms on that for all environments of the inequality

Meet for all. The space of analytic functionals carried on is therefore a distribution space. It is a subset of the space of distributions with compact support.

Definition

After Mikio Sato

A hyper function in one dimension is according to Sato by a pair of holomorphic functions, which are separated by a rim is shown. In most cases, a part of the real number axis. In this case, is defined in an open portion of the lower half of the complex plane, and in an open portion of the upper half of the complex plane. A hyperfunction is the " jump " from over the edge.

After André Martineau

Be an open and bounded subset. Then the space of hyperfunctions on is by

Defined.

Examples

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