Hyperfunction
المؤلف:
Isao, I.
المصدر:
Applied Hyperfunction Theory. Amsterdam, Netherlands: Kluwer, 1992.
الجزء والصفحة:
...
28-11-2018
995
Hyperfunction
A hyperfunction, discovered by Mikio Sato in 1958, is defined as a pair of holomorphic functions 
which are separated by a boundary 
. If 
is taken to be a segment on the real-line, then f is defined on the open region 
below the boundary and 
is defined on the open region 
above the boundary. A hyperfunction 
defined on gamma is the "jump" across the boundary from 
to 
.
This 
pair forms an equivalence class of pairs of holomorphic functions 
, where 
is a holomorphic function defined on the open region 
, comprised of both 
and 
.
Hyperfunctions can be shown to satisfy
There is no general product between hyperfunctions, but the product of a hyperfunction by a holomorphic function 
can be expressed as
 |
(3)
|
A standard holomorphic function 
can also be defined as a hyperfunction,
 |
(4)
|
The Heaviside step function 
and the delta function 
can be defined as the hyperfunctions
REFERENCES:
Isao, I. Applied Hyperfunction Theory. Amsterdam, Netherlands: Kluwer, 1992.
Penrose, R. The Road to Reality: A Complete Guide to the Laws of the Universe. New York: Random House, 2006.
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