Hyperfunction
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, building upon earlier work by Laurent Schwartz, Grothendieck and others.
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Formulation[edit]
A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the upper halfplane and another on the lower halfplane. That is, a hyperfunction is specified by a pair (f, g), where f is a holomorphic function on the upper halfplane and g is a holomorphic function on the lower halfplane.
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[edit]
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:
Concretely, let and be the upper halfplane and lower halfplane respectively. Then so
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.
Examples[edit]
 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
 The Dirac delta "function" is represented by

 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 nontrivial, 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

 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
 If f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e^{1/z}), then is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then is a distribution, so when f has an essential singularity then looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)
Operations on hyperfunctions[edit]
Let be any open subset.
 By definition is a vector space such that addition and multiplication with complex numbers are welldefined. Explicitly:
 The obvious restriction maps turn into a sheaf (which is in fact flabby).
 Multiplication with real analytic functions and differentiation are welldefined:

 With these definitions becomes a Dmodule and the embedding is a morphism of Dmodules.
 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 welldefined:

 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

 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 Lebesgueintegral. Furthermore: If is a distribution with compact support, is a real analytic function, and then
 Thus this notion of integration gives a precise meaning to formal expressions like
 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 welldefined operator from to :
See also[edit]
References[edit]
 Imai, Isao (2012) [1992], Applied Hyperfunction Theory, Mathematics and its Applications (Book 8), Springer, ISBN 9789401051255.
 Kaneko, Akira (1988), Introduction to the Theory of Hyperfunctions, Mathematics and its Applications (Book 3), Springer, ISBN 9789027728371
 Kashiwara, Masaki; Kawai, Takahiro; Kimura, Tatsuo (2017) [1986], Foundations of Algebraic Analysis, Princeton Legacy Library (Book 5158), PMS37, translated by Kato, Goro (Reprint ed.), Princeton University Press, ISBN 9780691628325
 Komatsu, Hikosaburo, ed. (1973), Hyperfunctions and PseudoDifferential Equations, Proceedings of a Conference at Katata, 1971, Lecture Notes in Mathematics 287, Springer, ISBN 9783540062189.
 Komatsu, Hikosaburo, Relative cohomology of sheaves of solutions of differential equations, pp. 192–261.
 Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki, Microfunctions and pseudodifferential eauations, pp. 265–529.  It is called SKK.
 Martineau, André (1960–1961), Les hyperfonctions de M. Sato, Séminaire Bourbaki, Tome 6 (19601961), 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 9780821845714.
 Pham, F. L., ed. (1975), Hyperfunctions and Theoretical Physics, Rencontre de Nice, 2130 Mai 1973, Lecture Notes in Mathematics 449, Springer, ISBN 9783540374541.
 Cerezo, A.; Piriou, A.; Chazarain, J., Introduction aux hyperfonctions, pp. 1–53.
 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 9783540049159.
 Schlichtkrull, Henrik (2013) [1984], Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progress in Mathematics (Softcover reprint of the original 1st ed.), Springer, ISBN 9781461297758
External links[edit]
 Jacobs, Bryan. "Hyperfunction". MathWorld.
 Kaneko, A. (2001) [1994], "Hyperfunction", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104