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Wikipedia / 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 distributionss of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958 in Japanese, (1959, 1960 in English), building upon earlier work by Laurent Schwartz, Grothendieck and others.

Formulation

A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the upper half-plane and another on the lower half-plane. That is, 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.

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 be the sheaf of holomorphic functions Define the hyperfunctions on the real line as the first local cohomology group:

be the upper half-plane and lower half-plane so

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.

is any open set giving another reason to think of hyperfunctions as "boundary values" of 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 Heaviside step function is the principal value of the complex logarithm of .

  • The Dirac delta "function"This is really a restatement of Cauchy's integral formula. To verify it one can calculate the integration of f just below the real line, and subtract integration of g just above the real line – both from left to right. Note that the hyperfunction can be non-trivial, even if the components are analytic continuation of the same function. Also this can be easily checked by differentiating the Heaviside function.

  • 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 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.

  • Using a partition of unity

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

Operations on hyperfunctions

be any open subset.

  • is a vector space

  • into a sheaf (which is in fact flabby).

  • becomes a D-module is a morphism of D-modules.

  • is called a holomorphic point The integrals are independent of the choice of these curves because the upper and lower half plane are simply connected.

  • be the space of hyperfunctions with compact support. Via the bilinear form This induces an identification of the dual space.

See also

References

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