Hyperfunction

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 be the sheaf of holomorphic functions on C and let C⁺ and C⁻
be the upper half plane and lower half plane respectively. Therefore

Define the hyperfunctions on the real line by

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 . 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

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.)
  

  Further reading

• Hörmander, Lars The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Springer-Verlag, Berlin, 2003. ISBN 3-540-00662-1