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.
𝒪 be the
sheaf of
holomorphic functions on ℂ+ and Hℝ1(ℂ,𝒪)=[H0(ℂ+,𝒪)⊕H0(ℂ−,𝒪)]/H0(ℂ,𝒪).
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 H0(U~∖U,𝒪)/H0(U~,𝒪) where U~ giving another reason to think of hyperfunctions as "boundary values" of holomorphic functions.
- 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 (12πiz,12πiz).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 (f,−f) is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then U⊆ℝ be any open subset.
- By definition ℬ into a sheaf (which is in fact flabby).
- Multiplication with real analytic functions ℬ(U) becomes a D-module and the embedding f∈ℬ(U) if a⩽b are two holomorphic points, then integration is well-defined: γ±(0)=a,γ±(1)=b. The integrals are independent of the choice of these curves because the upper and lower half plane are simply connected.
- Let 𝒪(U). This induces an identification of the dual space, Cc0(U) (or u∈ℰ′(U) is a distribution with compact support, ∫abu⋅φ=⟨u,φ⟩. Thus this notion of integration gives a precise meaning to formal expressions like 𝒪′(U). This is an alternative description of the same embedding ℝ, then composition with ℬ(U): Kashiwara, Masaki; Kawai, Takahiro; Kimura, Tatsuo (2017) [1986], Foundations of Algebraic Analysis, Princeton Legacy Library (Book 5158), vol. PMS-37, translated by Kato, Goro (Reprint ed.), Princeton University Press, ISBN 978-0-691-62832-5
- Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki, Microfunctions and pseudo-differential equations, pp. 265–529
- Sato, Mikio (1958), "Cyōkansū no riron (Theory of Hyperfunctions)", Sūgaku (in Japanese), 10 (1), Mathematical Society of Japan: 1–27, doi:10.11429/sugaku1947.10.1, ISSN 0039-470X
- Schapira, Pierre (1970), Theories des Hyperfonctions, Lecture Notes in Mathematics 126, Springer, ISBN 978-3-540-04915-9
- Jacobs, Bryan. "Hyperfunction". MathWorld.