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, building upon earlier work by Grothendieck and others.

Formulation
We want a hyperfunction on the real line to be the 'difference' between one holomorphic function on the upper half-plane and another on the lower half-plane. The easiest way to achieve this is to say that 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 f - g 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 $$\mathcal{O}$$ be the sheaf of holomorphic functions on C. Define the hyperfunctions on the real line by


 * $$\mathcal{B}(\mathbf{R}) = H^1_{\mathbf{R}}(\mathbf{C}, \mathcal{O}),$$

the first local cohomology group.

Concretely, let C+ and C&minus; be the upper half-plane and lower half-plane respectively. Then


 * $$\mathbf{C}^+ \cup \mathbf{C}^- = \mathbf{C} \setminus \mathbf{R}.\,$$

so


 * $$H^1_{\mathbf{R}}(\mathbf{C}, \mathcal{O}) = \left [ H^0(\mathbf{C}^+, \mathcal{O}) \oplus H^0(\mathbf{C}^-, \mathcal{O}) \right ] /H^0(\mathbf{C}, \mathcal{O}).$$

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.

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 (f, 0) or (0, -f).


 * The Dirac delta "function" is represented by $$\left(-\frac{1}{2\pi iz},-\frac{1}{2\pi iz}\right)$$. This is really a restatement of Cauchy's integral formula.


 * If g is a continuous function (or more generally a distribution) on the real line with support contained in a bounded interval I, then g corresponds to the hyperfunction (f, &minus;f), where f is a holomorphic function on the complement of I defined by


 * $$ f(z)={1\over 2\pi i}\int_{x\in I} g(x){dx\over z-x}.$$


 * This function f jumps in value by g(x) when crossing the real axis at the point x. The formula for f follows from the previous example by writing g as the convolution of itself with the Dirac delta function.


 * If f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e1/z), then (f, &minus;f) is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then (f, &minus;f) is a distribution, so when f has an essential singularity then (f,&minus;f) looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)