Definitions

# Hyperfunction

[hahy-per-fuhngk-shuhn]
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 (fg), 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 (fg) and (f+hg+h) are defined to be equivalent.

### Definition in one dimension

The motivation can be concretely implemented using ideas from sheaf cohomology. Let $mathcal\left\{O\right\}$ be the sheaf of holomorphic functions on C. Define the hyperfunctions on the real line by

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

the first local cohomology group.

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

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

so

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

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 $left\left(-frac\left\{1\right\}\left\{2pi iz\right\},-frac\left\{1\right\}\left\{2pi iz\right\}right\right)$. 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, −f), where f is a holomorphic function on the complement of I defined by

$f\left(z\right)=\left\{1over 2pi i\right\}int_\left\{xin I\right\} g\left(x\right)\left\{dxover z-x\right\}.$

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, −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.)

## References

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