Categories: Mathematical analysis | Complex analysis | Sheaf theory

Hyperfunction

In mathematics, hyperfunctions are sums of boundary values of holomorphic functions, and can be thought of informally as distributions of infinite order.

Contents

Motivation

We want the "boundary value" of a holomorphic function defined on the upper or lower half plane to be a hyperfunction on the real line. 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 lower half plane and g is a holomorphic function on the upper half plane. Informally, the hyperfunction (f, g) is the sum of the boundary values of f and g. If f is holomorphic on the whole complex plane, then it should have the same boundary values when considered as a function on either the upper or lower half plane. So (f, −f) should be considered to be 0. Similarly (f₁, g₁) and (f₂, g₂) represent the same hyperfunction if (and only if) f₁ − f₂ and g₂ − g₁ are restrictions of the same holomorphic function defined on the whole complex plane.

Formal definition

Let $\mathcal{O}$ be the sheaf of holomorphic functions on C and let C⁺ and C⁻ be the upper half plane and lower half plane respectively. Therefore

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

Then we have

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

Here, the left-hand side is the first sheaf cohomology group.

Define the hyperfunctions on the real line by

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

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 $(1 / z, - 1 / z) / 2π i$ . 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, −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, e^1/z), then (f, −f) is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then (f, −f) is a distribution, so when f has an essential singularity then (f,−f) looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)