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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

### Definition in one dimension

## Examples

## References

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.

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^{−}
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.

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

- $f(z)=\{1over\; 2pi\; i\}int\_\{xin\; I\}\; g(x)\{dxover\; z-x\}.$

- 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.)

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Last updated on Saturday June 28, 2008 at 04:23:45 PDT (GMT -0700)

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Last updated on Saturday June 28, 2008 at 04:23:45 PDT (GMT -0700)

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