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
distribution Distribution may refer to:
Mathematics
* Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a vari ...
s of infinite order. Hyperfunctions were introduced by
Mikio Sato in
1958
Events
January
* January 1 – The European Economic Community (EEC) comes into being.
* January 3 – The West Indies Federation is formed.
* January 4
** Edmund Hillary's Commonwealth Trans-Antarctic Expedition completes the third ...
in Japanese, (
1959
Events January
* January 1 - Cuba: Fulgencio Batista flees Havana when the forces of Fidel Castro advance.
* January 2 - Lunar probe Luna 1 was the first man-made object to attain escape velocity from Earth. It reached the vicinity of ...
,
1960
It is also known as the "Year of Africa" because of major events—particularly the independence of seventeen African nations—that focused global attention on the continent and intensified feelings of Pan-Africanism.
Events
January
* Jan ...
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
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds t ...
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
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
, 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
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* '' The Sheaf'', a student-run newspaper s ...
of
holomorphic functions on
Define the hyperfunctions on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
as the first
local cohomology group:
:
Concretely, let
and
be the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds t ...
and
lower half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
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
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
. 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 Distribution may refer to:
Mathematics
* Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a vari ...
) 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
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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, ''e''
1/''z''), then
is a hyperfunction with
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
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
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* '' The Sheaf'', a student-run newspaper s ...
(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
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunc ...
*
Generalized function
*
Distribution (mathematics)
*
Microlocal analysis
*
Pseudo-differential operator
*
Sheaf cohomology
References
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**. - It is called SKK.
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External links
*
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{{Authority control
Algebraic analysis
Complex analysis
Generalized functions
Sheaf theory