Hyperfunction

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 ${\displaystyle f-g}$ 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, the hyperfunctions (f, g) and (f + h, g + h) are defined to be equivalent.

• 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

${\displaystyle H(x)=\left(-{\tfrac {1}{2\pi i}}\log(z),-{\tfrac {1}{2\pi i}}\log(z)-1\right).}$

• The Dirac delta "function" is represented by

${\displaystyle \left({\tfrac {1}{2\pi iz}},{\tfrac {1}{2\pi iz}}\right).}$

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

${\displaystyle f(z)={\frac {1}{2\pi i}}\int _{x\in I}g(x){\frac {1}{z-x}}\,dx.}$

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 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 ${\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}$
• If f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e^1/z), then ${\displaystyle (f,-f)}$ is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then ${\displaystyle (f,-f)}$ is a distribution, so when f has an essential singularity then ${\displaystyle (f,-f)}$ looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)

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

• By definition ${\displaystyle {\mathcal {B}}(U)}$ is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly:

${\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}$

• The obvious restriction maps turn ${\displaystyle {\mathcal {B}}}$ into a sheaf (which is in fact flabby).
• Multiplication with real analytic functions ${\displaystyle h\in {\mathcal {O}}(U)}$ and differentiation are well-defined:

${\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\[6pt]{\frac {d}{dz}}(f_{+},f_{-})&:=\left({\frac {df_{+}}{dz}},{\frac {df_{-}}{dz}}\right)\end{aligned}}}$

With these definitions ${\displaystyle {\mathcal {B}}(U)}$ becomes a D-module and the embedding ${\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}}$ is a morphism of D-modules.

• A point ${\displaystyle a\in U}$ is called a holomorphic point of ${\displaystyle f\in {\mathcal {B}}(U)}$ if ${\displaystyle f}$ restricts to a real analytic function in some small neighbourhood of ${\displaystyle a.}$ If ${\displaystyle a\leqslant b}$ are two holomorphic points, then integration is well-defined:

${\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+}}f_{+}(z)\,dz+\int _{\gamma _{-}}f_{-}(z)\,dz}$

where ${\displaystyle \gamma _{\pm }:[0,1]\to \mathbb {C} ^{\pm }}$ are arbitrary curves with ${\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.}$ The integrals are independent of the choice of these curves because the upper and lower half plane are simply connected.

• Let ${\displaystyle {\mathcal {B}}_{c}(U)}$ be the space of hyperfunctions with compact support. Via the bilinear form

${\displaystyle {\begin{cases}{\mathcal {B}}_{c}(U)\times {\mathcal {O}}(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases}}}$

one associates to each hyperfunction with compact support a continuous linear function on ${\displaystyle {\mathcal {O}}(U).}$ This induces an identification of the dual space, ${\displaystyle {\mathcal {O}}'(U),}$ with ${\displaystyle {\mathcal {B}}_{c}(U).}$ A special case worth considering is the case of continuous functions or distributions with compact support: If one considers ${\displaystyle C_{c}^{0}(U)}$ (or ${\displaystyle {\mathcal {E}}'(U)}$) as a subset of ${\displaystyle {\mathcal {B}}(U)}$ via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If ${\displaystyle u\in {\mathcal {E}}'(U)}$ is a distribution with compact support, ${\displaystyle \varphi \in {\mathcal {O}}(U)}$ is a real analytic function, and ${\displaystyle \operatorname {supp} (u)\subset (a,b)}$ then

${\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .}$

Thus this notion of integration gives a precise meaning to formal expressions like

${\displaystyle \int _{a}^{b}\delta (x)\,dx}$

which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in ${\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)}$ is a subspace of ${\displaystyle {\mathcal {O}}'(U)}$. This is an alternative description of the same embedding ${\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}}$.

• If ${\displaystyle \Phi :U\to V}$ is a real analytic map between open sets of ${\displaystyle \mathbb {R} }$, then composition with ${\displaystyle \Phi }$ is a well-defined operator from ${\displaystyle {\mathcal {B}}(V)}$ to ${\displaystyle {\mathcal {B}}(U)}$:

${\displaystyle f\circ \Phi$ :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}

• Imai, Isao (2012) [1992], Applied Hyperfunction Theory, Mathematics and its Applications (Book 8), Springer, ISBN 978-94-010-5125-5.
• Kaneko, Akira (1988), Introduction to the Theory of Hyperfunctions, Mathematics and its Applications (Book 3), Springer, ISBN 978-90-277-2837-1
• Kashiwara, Masaki; Kawai, Takahiro; Kimura, Tatsuo (2017) [1986], Foundations of Algebraic Analysis, Princeton Legacy Library (Book 5158), PMS-37, translated by Kato, Goro (Reprint ed.), Princeton University Press, ISBN 978-0-691-62832-5
• Komatsu, Hikosaburo, ed. (1973), Hyperfunctions and Pseudo-Differential Equations, Proceedings of a Conference at Katata, 1971, Lecture Notes in Mathematics 287, Springer, ISBN 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. - It is called SKK.
• Martineau, André (1960–1961), Les hyperfonctions de M. Sato, Séminaire Bourbaki, Tome 6 (1960-1961), Exposé no. 214, MR 1611794, Zbl 0122.34902.
• Morimoto, Mitsuo (1993), An Introduction to Sato's Hyperfunctions, Translations of Mathematical Monographs (Book 129), American Mathematical Society, ISBN 978-0-82184571-4.
• Pham, F. L., ed. (1975), Hyperfunctions and Theoretical Physics, Rencontre de Nice, 21-30 Mai 1973, Lecture Notes in Mathematics 449, Springer, ISBN 978-3-540-37454-1.
  • Cerezo, A.; Piriou, A.; Chazarain, J., Introduction aux hyperfonctions, pp. 1–53.
• Sato, Mikio (1958), "Cyōkansū no riron (Theory of Hyperfunctions)", Sūgaku (in Japanese), Mathematical Society of Japan, 10 (1): 1–27, doi:10.11429/sugaku1947.10.1, 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:2261/6027, 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:2261/6031, MR 0132392.
• Schapira, Pierre (1970), Theories des Hyperfonctions, Lecture Notes in Mathematics 126, Springer, ISBN 978-3-540-04915-9.
• Schlichtkrull, Henrik (2013) [1984], Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progress in Mathematics (Softcover reprint of the original 1st ed.), Springer, ISBN 978-1-4612-9775-8

• Jacobs, Bryan. "Hyperfunction". MathWorld.
• Kaneko, A. (2001) [1994], "Hyperfunction", Encyclopedia of Mathematics, EMS Press