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{{Short description|Type of generalized function}}
{{about|hyperfunctions in a mathematical context|biological hypo- or hyperfunctions|endocrine disease}}
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 (mathematics)|distribution]]s of infinite order. Hyperfunctions were introduced by [[Mikio Sato]] in [[#CITEREFSato1958|1958]] in Japanese, ([[#CITEREFSato1959|1959]], [[#CITEREFSato1960|1960]] in English), building upon earlier work by [[Laurent Schwartz]], [[Alexander Grothendieck|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'',&nbsp;''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 <math>f -g</math> 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'',&nbsp;''g'') and (''f''&nbsp;+&nbsp;''h'',&nbsp;''g''&nbsp;+&nbsp;''h'') are defined to be equivalent.

===Definition in one dimension===
The motivation can be concretely implemented using ideas from [[sheaf cohomology]].  Let <math>\mathcal{O}</math> be the [[sheaf (mathematics)|sheaf]] of [[holomorphic function]]s on <math>\Complex.</math> Define the hyperfunctions on the [[real line]] as the first [[local cohomology]] group:

:<math>\mathcal{B}(\R) = H^1_{\R}(\Complex , \mathcal{O}).</math>

Concretely, let <math>\Complex^+</math> and <math>\Complex^-</math> be the [[upper half-plane]] and [[lower half-plane]] respectively. Then <math>\Complex^+ \cup \Complex^- = \Complex \setminus \R</math> so

:<math>H^1_{\R  }(\Complex , \mathcal{O}) = \left [ H^0(\Complex ^+, \mathcal{O}) \oplus H^0(\Complex ^-, \mathcal{O}) \right ] /H^0(\Complex , \mathcal{O}).</math>

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 <math>\mathcal{B}(U)</math> for any open set <math>U\subseteq\R</math> as the quotient <math>H^0(\tilde{U}\setminus U,\mathcal{O}) / H^0(\tilde{U},\mathcal{O})</math> where <math>\tilde{U}\subseteq\Complex</math> is any open set with <math>\tilde{U}\cap\mathbb{R}=U</math>. One can show that this definition does not depend on the choice of <math>\tilde{U}</math> 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'',&nbsp;0) or (0,&nbsp;−''f'').
*The [[Heaviside step function]] can be represented as <math display="block">H(x) = \left(1-\frac{1}{2\pi i}\log(z),-\frac{1}{2\pi i}\log(z)\right).</math> where <math>\log(z)</math> is the [[Complex logarithm#Principal value|principal value of the complex logarithm]] of {{mvar|z}}.
*The [[Dirac delta function|Dirac delta "function"]] is represented by <math display="block">\left(\dfrac{1}{2\pi iz}, \dfrac{1}{2\pi iz}\right).</math>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 (mathematics)|distribution]]) on the real line with support contained in a bounded interval ''I'', then ''g'' corresponds to the hyperfunction (''f'',&nbsp;−''f''), where ''f'' is a holomorphic function on the complement of ''I'' defined by <math display="block"> f(z)= \frac 1 {2\pi i} \int_{x\in I} g(x) \frac 1 {z-x} \, dx.</math> 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 <math>\textstyle\mathcal{D}'(\R)\to\mathcal{B}(\R).</math>
*If ''f'' is any function that is holomorphic everywhere except for an [[essential singularity]] at 0 (for example, ''e''<sup>1/''z''</sup>), then <math>(f, -f)</math> is a hyperfunction with [[Support (mathematics)|support]] 0 that is not a distribution. If ''f'' has a pole of finite order at 0 then <math>(f, -f)</math> is a distribution, so when ''f'' has an essential singularity then <math>(f, -f)</math> looks like a "distribution of infinite order" at 0. (Note that distributions always have ''finite'' order at any point.)

==Operations on hyperfunctions==

Let <math>U\subseteq\R</math> be any open subset.

* By definition <math>  \mathcal{B}(U)</math> is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly: <math display="block">a(f_+,f_-)+b(g_+,g_-) := (af_++bg_+, af_-+bg_-)</math>
* The obvious restriction maps turn <math> \mathcal{B}</math> into a [[sheaf (mathematics)|sheaf]] (which is in fact [[flabby sheaf|flabby]]).
* Multiplication with real analytic functions <math>h\in\mathcal{O}(U)</math> and differentiation are well-defined:<math display="block">\begin{align}
h(f_+,f_-) &:= (hf_+, hf_-) \\ [6pt]
\frac{d}{dz}(f_+,f_-) &:= \left (\frac{df_+}{dz},\frac{df_-}{dz} \right)
\end{align}</math> With these definitions <math>\mathcal{B}(U)</math> becomes a [[D-module]] and the embedding <math>\mathcal{D}'\hookrightarrow\mathcal{B}</math> is a morphism of D-modules.
* A point <math>a\in U</math> is called a ''holomorphic point'' of <math>f\in\mathcal{B}(U)</math> if <math>f</math> restricts to a real analytic function in some small neighbourhood of <math>a.</math> If <math>a\leqslant b</math> are two holomorphic points, then integration is well-defined: <math display="block">\int_a^b f := -\int_{\gamma_+} f_+(z) \, dz + \int_{\gamma_-} f_-(z) \, dz</math> where <math>\gamma_{\pm}:[0,1] \to \Complex^{\pm}</math> are arbitrary curves with <math>\gamma_{\pm}(0)=a, \gamma_{\pm}(1)=b.</math> The integrals are independent of the choice of these curves because the upper and lower half plane are [[simply connected]].
*Let <math>\mathcal{B}_c(U)</math> be the space of hyperfunctions with compact support. Via the bilinear form <math display="block">\begin{cases}
\mathcal{B}_c(U)\times\mathcal{O}(U)\to\Complex \\
(f,\varphi)\mapsto\int f \cdot \varphi
\end{cases}</math> one associates to each hyperfunction with compact support a continuous linear function on <math>\mathcal{O}(U).</math> This induces an identification of the dual space, <math>\mathcal{O}'(U),</math> with <math>\mathcal{B}_c(U).</math> A special case worth considering is the case of continuous functions or distributions with compact support: If one considers <math>C_c^0(U)</math> (or <math>\mathcal{E}'(U)</math>) as a subset of <math>\mathcal{B}(U)</math> via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If <math>u\in\mathcal{E}'(U)</math> is a distribution with compact support, <math>\varphi\in\mathcal{O}(U)</math> is a real analytic function, and <math>\operatorname{supp}(u)\subset(a,b)</math> then <math display="block">\int_a^b u\cdot\varphi = \langle u,\varphi\rangle.</math> Thus this notion of integration gives a precise meaning to formal expressions like <math display="block">\int_a^b \delta(x) \, dx</math> which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in <math>\mathcal{E}(U), \mathcal{E}'(U)</math> is a subspace of <math>\mathcal{O}'(U)</math>. This is an alternative description of the same embedding <math>\mathcal{E}'\hookrightarrow\mathcal{B}</math>.
* If <math>\Phi:U\to V</math> is a real analytic map between open sets of <math>\R</math>, then composition with <math>\Phi</math> is a well-defined operator from <math>\mathcal{B}(V)</math> to <math>\mathcal{B}(U)</math>: <math display="block">f\circ\Phi := (f_+\circ\Phi,f_-\circ\Phi)</math>

==See also==
{{Div col}}
*[[Algebraic analysis]]
*[[Generalized function]]
*[[Distribution (mathematics)]]
*[[Microlocal analysis]]
*[[Pseudo-differential operator]]
*[[Sheaf cohomology]]
{{Div col end}}

== References ==
*{{Citation
 |last=Imai
 |first=Isao
 |author-link=Isao Imai (physicist)
 |orig-year=1992
 |year=2012
 |title=Applied Hyperfunction Theory
 |series=Mathematics and its Applications (Book 8)
 |publisher=Springer
 |isbn=978-94-010-5125-5
 |url=https://www.springer.com/book/9780792315070
}}.
*{{Citation
 |last=Kaneko
 |first=Akira
 |date=1988
 |title=Introduction to the Theory of Hyperfunctions
 |publisher=Springer
 |series=Mathematics and its Applications (Japanese Series, Vol. 3)
 |isbn=978-90-277-2837-1
 |url=https://www.springer.com/book/9789027728371
}}
*{{Citation
 |last1=Kashiwara
 |first1=Masaki
 |author-link=Masaki Kashiwara
 |last2=Kawai
 |first2=Takahiro
 |last3=Kimura
 |first3=Tatsuo
 |translator=Kato, Goro
 |orig-year=1986
 |date=2017
 |title=Foundations of Algebraic Analysis
 |publisher=Princeton University Press
 |edition=Reprint
 |series=Princeton Legacy Library (Book 5158)
 |volume=PMS-37
 |isbn=978-0-691-62832-5
 |url=https://press.princeton.edu/titles/798.html
}}
*{{Citation
 |editor-last=Komatsu
 |editor-first=Hikosaburo
 |date=1973
 |title=Hyperfunctions and Pseudo-Differential Equations, Proceedings of a Conference at Katata, 1971
 |series=Lecture Notes in Mathematics 287
 |publisher=Springer
 |isbn=978-3-540-06218-9
 |url=https://www.springer.com/book/9783540062189
}}.
**{{Citation
 |last=Komatsu
 |first=Hikosaburo
 |title=Relative cohomology of sheaves of solutions of differential equations
 |pages=192–261
}}.
**{{Citation
 |last1=Sato
 |first1=Mikio
 |last2=Kawai
 |first2=Takahiro
 |last3=Kashiwara
 |first3=Masaki
 |title=Microfunctions and pseudo-differential equations
 |pages=265–529
}}. - It is called SKK.
*{{Citation
 |last=Martineau
 |first=André
 |author-link=André Martineau
 |date=1960–1961
 |title=Les hyperfonctions de M. Sato
 |series=Séminaire Bourbaki, Tome 6 (1960-1961), Exposé no. 214
 |mr=1611794
 |zbl=0122.34902
 |url=http://www.numdam.org/item/SB_1960-1961__6__127_0
}}.
*{{Citation
 |last=Morimoto
 |first=Mitsuo
 |date=1993
 |title=An Introduction to Sato's Hyperfunctions
 |series=Translations of Mathematical Monographs (Book 129)
 |publisher=American Mathematical Society
 |isbn=978-0-82184571-4
 |url=https://archive.org/details/introductiontosa0000mori
 |url-access=registration
 }}.
*{{Citation
 |editor-last=Pham
 |editor-first=F. L.
 |date=1975
 |title=Hyperfunctions and Theoretical Physics, Rencontre de Nice, 21-30 Mai 1973
 |series=Lecture Notes in Mathematics 449
 |publisher=Springer
 |isbn=978-3-540-37454-1
 |url=https://www.springer.com/book/9783540071518
}}.
**{{Citation
 |last1=Cerezo
 |first1=A.
 |last2=Piriou
 |first2=A.
 |last3=Chazarain
 |first3=J.
 |title=Introduction aux hyperfonctions
 |pages=1–53
}}.
*{{Citation
 |last=Sato
 |first=Mikio
 |author-link=Mikio Sato
 |year=1958
 |title=Cyōkansū no riron (Theory of Hyperfunctions)
 |publisher=Mathematical Society of Japan
 |journal=Sūgaku
 |volume=10
 |issue=1
 |pages=1–27
 |language=ja
 |issn=0039-470X
 |doi=10.11429/sugaku1947.10.1
 |url=https://www.jstage.jst.go.jp/article/sugaku1947/10/1/10_1_1/_article/-char/ja/
}}
* {{citation|last=Sato|first=Mikio|title=Theory of Hyperfunctions, I|hdl=2261/6027|journal=Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry|volume=8|year=1959|issue=1|pages=139–193|mr=0114124}}.
* {{citation|last=Sato|first=Mikio|title=Theory of Hyperfunctions, II|hdl=2261/6031|journal=Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry|volume=8|year=1960|issue=2|pages=387–437|mr=0132392}}.
*{{Citation
 |last=Schapira
 |first=Pierre
 |author-link=Pierre Schapira (mathematician)
 |date=1970
 |title=Theories des Hyperfonctions
 |series=Lecture Notes in Mathematics 126
 |publisher=Springer
 |isbn=978-3-540-04915-9
 |url=https://www.springer.com/book/9783540049159
}}.
*{{Citation
 |last=Schlichtkrull
 |first=Henrik 
 |orig-year=1984
 |date=2013
 |title=Hyperfunctions and Harmonic Analysis on Symmetric Spaces
 |publisher=Springer
 |edition=Softcover reprint of the original 1st
 |series=Progress in Mathematics
 |isbn=978-1-4612-9775-8
 |url=https://www.springer.com/book/9781461297758
}}

== External links ==
*{{MathWorld|title=Hyperfunction|urlname=Hyperfunction|author=Jacobs, Bryan}}
*{{SpringerEOM |id=Hyperfunction&oldid=16339 |title=Hyperfunction |author-first=A. |author-last=Kaneko }}

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[[Category:Algebraic analysis]]
[[Category:Complex analysis]]
[[Category:Generalized functions]]
[[Category:Sheaf theory]]