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

			(1)
			(2)

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

			(5)
			(6)

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.