# Hyperfunction

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Type of generalized function

This article is about hyperfunctions in a mathematical context. For biological hypo- or hyperfunctions, see [endocrine disease](/source/Endocrine_disease).

In [mathematics](/source/Mathematics), **hyperfunctions** are generalizations of functions, as a 'jump' from one [holomorphic function](/source/Holomorphic_function) to another at a boundary, and can be thought of informally as [distributions](/source/Distribution_(mathematics)) of infinite order. Hyperfunctions were introduced by [Mikio Sato](/source/Mikio_Sato) in [1958](#CITEREFSato1958) in Japanese, ([1959](#CITEREFSato1959), [1960](#CITEREFSato1960) in English), building upon earlier work by [Laurent Schwartz](/source/Laurent_Schwartz), [Grothendieck](/source/Alexander_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](/source/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 f − g {\displaystyle f-g} 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](/source/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](/source/Sheaf_cohomology). Let O {\displaystyle {\mathcal {O}}} be the [sheaf](/source/Sheaf_(mathematics)) of [holomorphic functions](/source/Holomorphic_function) on C . {\displaystyle \mathbb {C} .} Define the hyperfunctions on the [real line](/source/Real_line) as the first [local cohomology](/source/Local_cohomology) group:

- B ( R ) = H R 1 ( C , O ) . {\displaystyle {\mathcal {B}}(\mathbb {R} )=H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}}).}

Concretely, let C + {\displaystyle \mathbb {C} ^{+}} and C − {\displaystyle \mathbb {C} ^{-}} be the [upper half-plane](/source/Upper_half-plane) and [lower half-plane](/source/Lower_half-plane) respectively. Then C + ∪ C − = C ∖ R {\displaystyle \mathbb {C} ^{+}\cup \mathbb {C} ^{-}=\mathbb {C} \setminus \mathbb {R} } so

- H R 1 ( C , O ) = [ H 0 ( C + , O ) ⊕ H 0 ( C − , O ) ] / H 0 ( C , O ) . {\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}}).}

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 B ( U ) {\displaystyle {\mathcal {B}}(U)} for any open set U ⊆ R {\displaystyle U\subseteq \mathbb {R} } as the quotient H 0 ( U ~ ∖ U , O ) / H 0 ( U ~ , O ) {\displaystyle H^{0}({\tilde {U}}\setminus U,{\mathcal {O}})/H^{0}({\tilde {U}},{\mathcal {O}})} where U ~ ⊆ C {\displaystyle {\tilde {U}}\subseteq \mathbb {C} } is any [open set](/source/Open_set) with U ~ ∩ R = U {\displaystyle {\tilde {U}}\cap \mathbb {R} =U} . One can show that this definition does not depend on the choice of U ~ {\displaystyle {\tilde {U}}} 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](/source/Heaviside_step_function) can be represented as H ( x ) = ( 1 − 1 2 π i log ⁡ ( z ) , − 1 2 π i log ⁡ ( z ) ) . {\displaystyle H(x)=\left(1-{\frac {1}{2\pi i}}\log(z),-{\frac {1}{2\pi i}}\log(z)\right).} where log ⁡ ( z ) {\displaystyle \log(z)} is the [principal value of the complex logarithm](/source/Complex_logarithm#Principal_value) of z.

- The [Dirac delta "function"](/source/Dirac_delta_function) is represented by ( 1 2 π i z , 1 2 π i z ) . {\displaystyle \left({\dfrac {1}{2\pi iz}},{\dfrac {1}{2\pi iz}}\right).} This is really a restatement of [Cauchy's integral formula](/source/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](/source/Analytic_continuation) of the same function. Also this can be easily checked by differentiating the Heaviside function.

- If *g* is a [continuous function](/source/Continuous_function) (or more generally a [distribution](/source/Distribution_(mathematics))) 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 ( z ) = 1 2 π i ∫ x ∈ I g ( x ) 1 z − x d x . {\displaystyle f(z)={\frac {1}{2\pi i}}\int _{x\in I}g(x){\frac {1}{z-x}}\,dx.} 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](/source/Convolution) of itself with the Dirac delta function.

- Using a [partition of unity](/source/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 D ′ ( R ) → B ( R ) . {\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}

- If *f* is any function that is holomorphic everywhere except for an [essential singularity](/source/Essential_singularity) at 0 (for example, *e*1/*z*), then ( f , − f ) {\displaystyle (f,-f)} is a hyperfunction with [support](/source/Support_(mathematics)) 0 that is not a distribution. If *f* has a pole of finite order at 0 then ( f , − f ) {\displaystyle (f,-f)} is a distribution, so when *f* has an essential singularity then ( f , − f ) {\displaystyle (f,-f)} looks like a "distribution of infinite order" at 0. (Note that distributions always have *finite* order at any point.)

## Operations on hyperfunctions

Let U ⊆ R {\displaystyle U\subseteq \mathbb {R} } be any open subset.

- By definition B ( U ) {\displaystyle {\mathcal {B}}(U)} is a [vector space](/source/Vector_space) such that addition and multiplication with complex numbers are well-defined. Explicitly: a ( f + , f − ) + b ( g + , g − ) := ( a f + + b g + , a f − + b g − ) {\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}

- The obvious restriction maps turn B {\displaystyle {\mathcal {B}}} into a [sheaf](/source/Sheaf_(mathematics)) (which is in fact [flabby](/source/Flabby_sheaf)).

- Multiplication with real analytic functions h ∈ O ( U ) {\displaystyle h\in {\mathcal {O}}(U)} and differentiation are well-defined: h ( f + , f − ) := ( h f + , h f − ) d d z ( f + , f − ) := ( d f + d z , d f − d z ) {\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\[6pt]{\frac {d}{dz}}(f_{+},f_{-})&:=\left({\frac {df_{+}}{dz}},{\frac {df_{-}}{dz}}\right)\end{aligned}}} With these definitions B ( U ) {\displaystyle {\mathcal {B}}(U)} becomes a [D-module](/source/D-module) and the embedding D ′ ↪ B {\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}} is a morphism of D-modules.

- A point a ∈ U {\displaystyle a\in U} is called a *holomorphic point* of f ∈ B ( U ) {\displaystyle f\in {\mathcal {B}}(U)} if f {\displaystyle f} restricts to a real analytic function in some small neighbourhood of a . {\displaystyle a.} If a ⩽ b {\displaystyle a\leqslant b} are two holomorphic points, then integration is well-defined: ∫ a b f := − ∫ γ + f + ( z ) d z + ∫ γ − f − ( z ) d z {\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+}}f_{+}(z)\,dz+\int _{\gamma _{-}}f_{-}(z)\,dz} where γ ± : [ 0 , 1 ] → C ± {\displaystyle \gamma _{\pm }:[0,1]\to \mathbb {C} ^{\pm }} are arbitrary curves with γ ± ( 0 ) = a , γ ± ( 1 ) = b . {\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.} The integrals are independent of the choice of these curves because the upper and lower half plane are [simply connected](/source/Simply_connected).

- Let B c ( U ) {\displaystyle {\mathcal {B}}_{\text{c}}(U)} be the space of hyperfunctions with compact support. Via the [bilinear form](/source/Bilinear_form) { B c ( U ) × O ( U ) → C ( f , φ ) ↦ ∫ f ⋅ φ {\displaystyle {\begin{cases}{\mathcal {B}}_{\text{c}}(U)\times {\mathcal {O}}(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases}}} one associates to each hyperfunction with compact support a continuous linear function on O ( U ) . {\displaystyle {\mathcal {O}}(U).} This induces an identification of the [dual space](/source/Dual_space), O ′ ( U ) , {\displaystyle {\mathcal {O}}'(U),} with B c ( U ) . {\displaystyle {\mathcal {B}}_{\text{c}}(U).} A special case worth considering is the case of continuous functions or distributions with compact support: If one considers C c 0 ( U ) {\displaystyle C_{\text{c}}^{0}(U)} (or E ′ ( U ) {\displaystyle {\mathcal {E}}'(U)} ) as a subset of B ( U ) {\displaystyle {\mathcal {B}}(U)} via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If u ∈ E ′ ( U ) {\displaystyle u\in {\mathcal {E}}'(U)} is a distribution with compact support, φ ∈ O ( U ) {\displaystyle \varphi \in {\mathcal {O}}(U)} is a real analytic function, and supp ⁡ ( u ) ⊂ ( a , b ) {\displaystyle \operatorname {supp} (u)\subset (a,b)} then ∫ a b u ⋅ φ = ⟨ u , φ ⟩ . {\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .} Thus this notion of integration gives a precise meaning to formal expressions like ∫ a b δ ( x ) d x {\displaystyle \int _{a}^{b}\delta (x)\,dx} which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in E ( U ) , E ′ ( U ) {\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)} is a subspace of O ′ ( U ) {\displaystyle {\mathcal {O}}'(U)} . This is an alternative description of the same embedding E ′ ↪ B {\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}} .

- If Φ : U → V {\displaystyle \Phi :U\to V} is a real analytic map between open sets of R {\displaystyle \mathbb {R} } , then composition with Φ {\displaystyle \Phi } is a well-defined operator from B ( V ) {\displaystyle {\mathcal {B}}(V)} to B ( U ) {\displaystyle {\mathcal {B}}(U)} : f ∘ Φ := ( f + ∘ Φ , f − ∘ Φ ) {\displaystyle f\circ \Phi :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}

## See also

- [Algebraic analysis](/source/Algebraic_analysis)

- [Generalized function](/source/Generalized_function)

- [Distribution (mathematics)](/source/Distribution_(mathematics))

- [Microlocal analysis](/source/Microlocal_analysis)

- [Pseudo-differential operator](/source/Pseudo-differential_operator)

- [Sheaf cohomology](/source/Sheaf_cohomology)

## References

- [Imai, Isao](/source/Isao_Imai_(physicist)) (2012) [1992], [*Applied Hyperfunction Theory*](https://www.springer.com/book/9780792315070), Mathematics and its Applications (Book 8), Springer, [ISBN](/source/ISBN_(identifier)) [978-94-010-5125-5](https://en.wikipedia.org/wiki/Special:BookSources/978-94-010-5125-5).

- Kaneko, Akira (1988), [*Introduction to the Theory of Hyperfunctions*](https://www.springer.com/book/9789027728371), Mathematics and its Applications (Japanese Series, Vol. 3), Springer, [ISBN](/source/ISBN_(identifier)) [978-90-277-2837-1](https://en.wikipedia.org/wiki/Special:BookSources/978-90-277-2837-1)

- [Kashiwara, Masaki](/source/Masaki_Kashiwara); Kawai, Takahiro; Kimura, Tatsuo (2017) [1986], [*Foundations of Algebraic Analysis*](https://press.princeton.edu/titles/798.html), Princeton Legacy Library (Book 5158), vol. PMS-37, translated by Kato, Goro (Reprint ed.), Princeton University Press, [ISBN](/source/ISBN_(identifier)) [978-0-691-62832-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-62832-5)

- Komatsu, Hikosaburo, ed. (1973), [*Hyperfunctions and Pseudo-Differential Equations, Proceedings of a Conference at Katata, 1971*](https://www.springer.com/book/9783540062189), Lecture Notes in Mathematics 287, Springer, [ISBN](/source/ISBN_(identifier)) [978-3-540-06218-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-06218-9) - Komatsu, Hikosaburo, *Relative cohomology of sheaves of solutions of differential equations*, pp. 192–261. - Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki, *Microfunctions and pseudo-differential equations*, pp. 265–529

- [Martineau, André](/source/Andr%C3%A9_Martineau) (1960–1961), [*Les hyperfonctions de M. Sato*](http://www.numdam.org/item/SB_1960-1961__6__127_0), Séminaire Bourbaki, Tome 6 (1960-1961), Exposé no. 214, [MR](/source/MR_(identifier)) [1611794](https://mathscinet.ams.org/mathscinet-getitem?mr=1611794), [Zbl](/source/Zbl_(identifier)) [0122.34902](https://zbmath.org/?format=complete&q=an:0122.34902)

- Morimoto, Mitsuo (1993), [*An Introduction to Sato's Hyperfunctions*](https://archive.org/details/introductiontosa0000mori), Translations of Mathematical Monographs (Book 129), American Mathematical Society, [ISBN](/source/ISBN_(identifier)) [978-0-82184571-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-82184571-4)

- Pham, F. L., ed. (1975), [*Hyperfunctions and Theoretical Physics, Rencontre de Nice, 21-30 Mai 1973*](https://www.springer.com/book/9783540071518), Lecture Notes in Mathematics 449, Springer, [ISBN](/source/ISBN_(identifier)) [978-3-540-37454-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-37454-1) - Cerezo, A.; Piriou, A.; Chazarain, J., *Introduction aux hyperfonctions*, pp. 1–53

- [Sato, Mikio](/source/Mikio_Sato) (1958), ["Cyōkansū no riron (Theory of Hyperfunctions)"](https://www.jstage.jst.go.jp/article/sugaku1947/10/1/10_1_1/_article/-char/ja/), *Sūgaku* (in Japanese), **10** (1), Mathematical Society of Japan: 1–27, [doi](/source/Doi_(identifier)):[10.11429/sugaku1947.10.1](https://doi.org/10.11429%2Fsugaku1947.10.1), [ISSN](/source/ISSN_(identifier)) [0039-470X](https://search.worldcat.org/issn/0039-470X)

- Sato, Mikio (1959), "Theory of Hyperfunctions, I", *Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry*, **8** (1): 139–193, [hdl](/source/Hdl_(identifier)):[2261/6027](https://hdl.handle.net/2261%2F6027), [MR](/source/MR_(identifier)) [0114124](https://mathscinet.ams.org/mathscinet-getitem?mr=0114124)

- Sato, Mikio (1960), "Theory of Hyperfunctions, II", *Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry*, **8** (2): 387–437, [hdl](/source/Hdl_(identifier)):[2261/6031](https://hdl.handle.net/2261%2F6031), [MR](/source/MR_(identifier)) [0132392](https://mathscinet.ams.org/mathscinet-getitem?mr=0132392)

- [Schapira, Pierre](/source/Pierre_Schapira_(mathematician)) (1970), [*Theories des Hyperfonctions*](https://www.springer.com/book/9783540049159), Lecture Notes in Mathematics 126, Springer, [ISBN](/source/ISBN_(identifier)) [978-3-540-04915-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-04915-9)

- Schlichtkrull, Henrik (2013) [1984], [*Hyperfunctions and Harmonic Analysis on Symmetric Spaces*](https://www.springer.com/book/9781461297758), Progress in Mathematics (Softcover reprint of the original 1st ed.), Springer, [ISBN](/source/ISBN_(identifier)) [978-1-4612-9775-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-9775-8)

## External links

- Jacobs, Bryan. ["Hyperfunction"](https://mathworld.wolfram.com/Hyperfunction.html). *[MathWorld](/source/MathWorld)*.

- Kaneko, A. (2001) [1994], ["Hyperfunction"](https://www.encyclopediaofmath.org/index.php?title=Hyperfunction), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society)

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