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 distributions 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. Informally, the hyperfunction is what the difference  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. Let  be the sheaf of holomorphic functions on  Define the hyperfunctions on the real line as the first local cohomology group:

Concretely, let  and  be the upper half-plane and lower half-plane respectively. Then  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. More generally one can define  for any open set  as the quotient  where  is any open set with . One can show that this definition does not depend on the choice of  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 can be represented as  where  is the principal value of the complex logarithm of . The Dirac delta "function" is represented by 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 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. Using a partition of unity one can write any continuous function (distribution) as a locally finite sum of functions (distributions) with compact support. This can be exploited to extend the above embedding to an embedding 
If f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e1/z), then  is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then  is a distribution, so when f has an essential singularity then  looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.) Operations on hyperfunctions

Let  be any open subset. By definition  is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly: 
 The obvious restriction maps turn  into a sheaf (which is in fact flabby). Multiplication with real analytic functions  and differentiation are well-defined: With these definitions  becomes a D-module and the embedding  is a morphism of D-modules. A point  is called a holomorphic point of  if  restricts to a real analytic function in some small neighbourhood of  If  are two holomorphic points, then integration is well-defined:  where  are arbitrary curves with  The integrals are independent of the choice of these curves because the upper and lower half plane are simply connected. Let  be the space of hyperfunctions with compact support. Via the bilinear form  one associates to each hyperfunction with compact support a continuous linear function on  This induces an identification of the dual space,  with  A special case worth considering is the case of continuous functions or distributions with compact support: If one considers  (or ) as a subset of  via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If  is a distribution with compact support,  is a real analytic function, and  then  Thus this notion of integration gives a precise meaning to formal expressions like  which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in  is a subspace of . This is an alternative description of the same embedding . If  is a real analytic map between open sets of , then composition with  is a well-defined operator from  to :

See also

Algebraic analysis
Generalized function
Distribution (mathematics)
Microlocal analysis
Pseudo-differential operator
Sheaf cohomology

References 
. . . . - It is called SKK. . . . . . . . External links 

Algebraic analysis
Complex analysis
Generalized functions
Sheaf theory