In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in [|1958] in Japanese,, building upon earlier work by Laurent Schwartz, Grothendieck and others.
Formulation
A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the upper half-plane and another on the lower half-plane. That is, a hyperfunction is specified by a pair, where f is a holomorphic function on the upper half-plane and g is a holomorphic function on the lower half-plane. Informally, the hyperfunction is what the difference would be at the real line itself. This difference is not affected by adding the same holomorphic function to both f and g, so if h is a holomorphic function on the whole complex plane, the hyperfunctions and are defined to be equivalent.
The motivation can be concretely implemented using ideas from sheaf cohomology. Let be the sheaf of holomorphic functions on Define the hyperfunctions on the real line as the first local cohomology group: Concretely, let and be the upper half-plane and lower half-plane respectively. Then so Since the zeroth cohomology group of any sheaf is simply the global sections of that sheaf, we see that a hyperfunction is a pair of holomorphic functions one each on the upper and lower complex halfplane modulo entire holomorphic functions. More generally one can define for any open set as the quotient where is any open set with. One can show that this definition does not depend on the choice of giving another reason to think of hyperfunctions as "boundary values" of holomorphic functions.
Examples
If f is any holomorphic function on the whole complex plane, then the restriction of f to the real axis is a hyperfunction, represented by either or.
If g is a continuous function on the real line with support contained in a bounded intervalI, then g corresponds to the hyperfunction, where f is a holomorphic function on the complement of I defined by
Using a partition of unity one can write any continuous function as a locally finite sum of functions with compact support. This can be exploited to extend the above embedding to an embedding
If f is any function that is holomorphic everywhere except for an essential singularity at 0, then is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then is a distribution, so when f has an essential singularity then looks like a "distribution of infinite order" at 0.
Multiplication with real analytic functions and differentiation are well-defined:
A point is called a holomorphic point of if restricts to a real analytic function in some small neighbourhood of If are two holomorphic points, then integration is well-defined:
