Type of generalized function
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:
![{\displaystyle {\mathcal {B))(\mathbb {R} )=H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O))).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/534a9bd483c93e16d6e21cd087bdc9a079c1c538)
Concretely, let
and
be the upper half-plane and lower half-plane respectively. Then
so
![{\displaystyle H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O)))=\left[H^{0}(\mathbb {C} ^{+},{\mathcal {O)))\oplus H^{0}(\mathbb {C} ^{-},{\mathcal {O)))\right]/H^{0}(\mathbb {C} ,{\mathcal {O))).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/973e93046c1d3bda18b6090c96598cde598a26a1)
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.
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: ![{\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fea250e16b9896b9982dd86099ea5b3557ad5a52)
- The obvious restriction maps turn
into a sheaf (which is in fact flabby).
- Multiplication with real analytic functions
and differentiation are well-defined:![{\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\[6pt]{\frac {d}{dz))(f_{+},f_{-})&:=\left({\frac {df_{+)){dz)),{\frac {df_{-)){dz))\right)\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a14e67df7ebbd7d256cacbc70f335d997da735c)
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: ![{\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+))f_{+}(z)\,dz+\int _{\gamma _{-))f_{-}(z)\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6270edb9019db7ba6ede10a7a97215d7891067c3)
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 ![{\displaystyle {\begin{cases}{\mathcal {B))_{c}(U)\times {\mathcal {O))(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f74abf3a0021960fd2134276d5e1a5a685931cd)
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 ![{\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1187ce487751eda1b148ebe518b8e1afdf854126)
Thus this notion of integration gives a precise meaning to formal expressions like ![{\displaystyle \int _{a}^{b}\delta (x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7cd7ace0993c50ad6baf623977785ab94d4dd5)
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
: ![{\displaystyle f\circ \Phi :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd50d2072c33f25c7afc6a27733454d6254fb66d)