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 : ${\displaystyle h(x)}$ ${\displaystyle f(x)}$ ${\displaystyle g(x)}$ ${\displaystyle \gamma }$

${\displaystyle h(x)=\left(\!|f(x),g(x)|\!\right)}$

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. ${\displaystyle (F_{+},F_{-})}$ ${\displaystyle F_{\pm }}$ ${\displaystyle \operatorname {Im} (z)>0}$ ${\displaystyle \operatorname {Im} (z)<0}$ ${\displaystyle (F,-F)}$ ${\displaystyle F}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

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 ${\displaystyle K\subset \mathbb {R} ^{n}}$ ${\displaystyle A(K)}$ ${\displaystyle f\colon K\to \mathbb {C} }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle A'(K)}$ ${\displaystyle K}$ ${\displaystyle u}$ ${\displaystyle A(\mathbb {R} ^{n})}$ ${\displaystyle \omega }$ ${\displaystyle K}$

${\displaystyle |u(\phi )|\leq C_{\omega }\sup _{\omega }|\phi |}$

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 . ${\displaystyle \phi \in A(\mathbb {R} ^{n})}$ ${\displaystyle A'(K)}$ ${\displaystyle K}$ ${\displaystyle {\mathcal {E}}(\mathbb {R} ^{n})}$ ${\displaystyle A(K)\subseteq {\mathcal {E}}(\mathbb {R} ^{n})}$ ${\displaystyle \{u\in {\mathcal {E}}'(\mathbb {R} ^{n})\mid \operatorname {supp} (u)\subseteq K\}}$ ${\displaystyle A'(K)}$

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 . ${\displaystyle (f,g)}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma }$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle \gamma }$

After André Martineau

Be an open and bounded subset. Then the space of Hyper functions on by ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle B(X)}$ ${\displaystyle X}$

${\displaystyle B(X):=A'({\bar {X}})/A'(\partial X)}$

Are defined.

Examples

Heaviside jump function

${\displaystyle \theta (x):=\left(\!\!\left|{\frac {1}{2\,\pi \,\mathrm {i} }}\,\log z,{\frac {1}{2\,\pi \,\mathrm {i} }}\,\log z-1\right|\!\!\right)}$

Dirac-Heaviside delta function

${\displaystyle \delta (x):={\frac {\mathrm {d} \theta (x)}{\mathrm {d} x}}=\left(\!\!\left|{\frac {1}{2\,\pi \,\mathrm {i} \,z}},{\frac {1}{2\,\pi \,\mathrm {i} \,z}}\right|\!\!\right)}$

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).