Hyperfunction

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, (1959, 1960 in English), 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 ("f", "g"), 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 formula_1 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 ("f", "g") and ("f" + "h", "g" + "h") are defined to be equivalent.
Definition in one dimension.
The motivation can be concretely implemented using ideas from sheaf cohomology. Let formula_2 be the sheaf of holomorphic functions on formula_3 Define the hyperfunctions on the real line as the first local cohomology group:
Concretely, let formula_5 and formula_6 be the upper half-plane and lower half-plane respectively. Then formula_7 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 formula_9 for any open set formula_10 as the quotient formula_11 where formula_12 is any open set with formula_13. One can show that this definition does not depend on the choice of formula_14 giving another reason to think of hyperfunctions as "boundary values" of holomorphic functions.
Examples.
where formula_16 is the principal value of the complex logarithm of .
Operations on hyperfunctions.
Let formula_10 be any open subset.