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A hyperfunction is a kind of [[Generalized function|generalized function]], and the class of hyperfunctions is characterized as the largest class of localizable generalized functions. According to the three general recipes for constructing generalized functions, hyperfunctions are obtained 1) as ideal limits or boundary values on the real axis of holomorphic functions; 2) as locally finite sums of continuous linear functionals on real-analytic functions; or 3) as formal derivatives of continuous functions by infinite-order differential operators of local type. The theory of hyperfunctions has a prehistory starting with G. Köthe, A. Grothendieck, etc., and also in some physical literature. But the localization together with the accurate setting in the case of several variables was given by M. Sato, who employed [[Cohomology|cohomology]] theory.

A hyperfunction of one variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484201.png" /> on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484202.png" /> is a formal expression of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484204.png" /> is a function holomorphic on the upper, respectively lower, half-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484205.png" />, for a complex neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484206.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484207.png" />. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484208.png" /> is identified with 0 if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h0484209.png" /> agrees on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842010.png" /> as a holomorphic function (or, equivalently, as a continuous function, by the [[Painlevé theorem|Painlevé theorem]]). If the limits exist in distribution sense, the formula gives the natural imbedding of the space of distributions into that of hyperfunctions. The precise definition of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842011.png" /> of hyperfunctions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842012.png" /> is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842013.png" />. (Actually the limit is superfluous.) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842014.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842015.png" /> denote the hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842017.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842018.png" />. In turn, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842019.png" /> is called a defining function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842020.png" />. Examples of typical generalized functions interpreted as hyperfunctions are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842021.png" /></td> </tr></table>

(Dirac's delta-function);

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842022.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842023.png" /></td> </tr></table>

(Heaviside's function);

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842024.png" /></td> </tr></table>

(Cauchy's principal value);

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842025.png" /></td> </tr></table>

(Hadamard's finite part);

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842026.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842027.png" /></td> </tr></table>

Hyperfunctions of several variables. In the sequel a cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842028.png" /> means an open [[Convex cone|convex cone]] with vertex at the origin. For two cones <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842029.png" />, the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842030.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842031.png" /> is relatively compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842032.png" />. A wedge means an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842033.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842035.png" /> is a real open set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842036.png" /> is a cone. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842037.png" /> is called the edge of the wedge and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842038.png" /> the opening. An infinitesimal wedge (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842040.png" />-wedge) or a tuboid of opening <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842041.png" /> and edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842042.png" /> (or simply of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842044.png" />) is a complex open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842046.png" /> and such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842047.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842048.png" /> contains the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842049.png" /> contained in some complex neighbourhood of the edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842050.png" />. The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842051.png" /> denotes any one such open set, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842052.png" /> denotes the totality of functions holomorphic on some of them (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842053.png" />, the limit being taken with respect to all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842054.png" />-wedges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842055.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842056.png" />).

A hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842057.png" /> on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842058.png" /> is an equivalence class (in the obvious sense) of formal expressions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842059.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842060.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842061.png" /> is called a set of defining functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842062.png" />. The totality of hyperfunctions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842063.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842064.png" />. It becomes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842065.png" />-linear space by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842066.png" />-linear structure of the defining functions. Linear differential operators with real-analytic coefficients act in the same way via the defining functions. The above boundary value representation corresponds to the case of one variable when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842068.png" />.

Hyperfunctions are localizable, i.e. the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842069.png" /> satisfies the axioms for a [[Sheaf|sheaf]]. It is a flabby sheaf, i.e. any section extends to the whole space. The sheaf of hyperfunctions is defined via the derived sheaf as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842071.png" /> is the orientation sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842072.png" />. Actually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842073.png" /> holds, the cohomologies in the other degrees being trivial. By choosing a Stein neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842074.png" />, one can represent the relative cohomology by a covering: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842077.png" />, then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842079.png" /> constitutes a relative Stein covering of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842080.png" />, hence by Leray's theorem,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842081.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842082.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842083.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842085.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842086.png" /> this reduces to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842087.png" />. Another representation is given by choosing a set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842088.png" /> such that the half-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842089.png" /> defined by them satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842090.png" />, and putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842092.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842093.png" />-cocycles are of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842094.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842096.png" /> is given as the quotient space of these by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842097.png" />-coboundary space. (The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842098.png" /> implies that the factor below it is omitted.)

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h04842099.png" /> is identified with the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420100.png" /> by extending the following correspondences:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420101.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420102.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420103.png" />-th orthant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420105.png" />; or by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420106.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420107.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420108.png" />.

Several practical criteria to determine whether a hyperfunction is zero are collectively known as the edge-of-the-wedge theorem: A hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420109.png" /> with single term is zero if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420110.png" /> itself is zero (injectivity of the boundary value operation). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420111.png" /> if and only if they stick together to a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420112.png" /> (an Epstein-type theorem). In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420113.png" /> the result becomes real analytic (a Bogolyubov-type theorem). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420114.png" /> if and only if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420116.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420118.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420120.png" /> (a Martineau-type theorem). These are manifestations of the fact that the cohomology classes vanish by covering representations.

A real-analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420121.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420122.png" /> is naturally included in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420123.png" /> by identifying it with the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420124.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420125.png" />. They form a subsheaf. Thus, for a hyperfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420126.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420127.png" /> the notion of the singular support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420128.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420129.png" />, is well defined: It is the complement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420130.png" /> of the largest open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420131.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420132.png" /> is real analytic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420133.png" /> for a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420134.png" />, then one can choose a boundary value representation such that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420135.png" /> can be continued analytically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420136.png" />. The edge-of-the-wedge theorem supplies also criteria for determining whether or not a hyperfunction is real analytic in an open (sub)set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420137.png" />.

The definite integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420138.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420139.png" /> over a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420140.png" /> is well-defined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420141.png" />. Indeed, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420142.png" /> be a boundary value representation such that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420143.png" /> may be extended to a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420144.png" />, or, more generally, to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420145.png" />-wedge with edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420146.png" /> and with an opening whose projection to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420147.png" />-space is the whole space. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420148.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420149.png" /> is a suitable deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420150.png" /> fixing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420151.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420152.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420153.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420154.png" />-plane. This integration commutes with differentiation or integration with respect to the remaining variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420155.png" />. Cf. [[Microlocal analysis|Microlocal analysis]] for further operations.

The totality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420156.png" /> of hyperfunctions with support in a fixed compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420157.png" /> becomes a nuclear [[Fréchet space|Fréchet space]] (cf. also [[Nuclear space|Nuclear space]]). It is the strong dual of the nuclear (DF)-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420158.png" /> of real-analytic functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420159.png" /> and endowed with the inductive limit topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420160.png" />'s for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420161.png" />. Namely, hyperfunctions with compact supports are analytic functionals with carrier in the real axis. The duality is given by the definite integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420162.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420164.png" />. Analytic functionals can be employed to reconstruct the space or the sheaf of hyperfunctions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420165.png" /> is  "the totality of locally finite sums of analytic functionals with carrier in W" , where rearrangement by decomposition of supports is admitted as an equivalence relation (Martineau's definition); or b) for a bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420166.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420167.png" /> (Schapira's definition).

The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420168.png" /> of Fourier hyperfunctions is defined on the directional compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420169.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420170.png" />, employing instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420171.png" /> the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420172.png" /> of holomorphic functions of infra-exponential growth, i.e. of growth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420173.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420174.png" />. Fourier hyperfunctions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420175.png" /> admit a Fourier transform of Bochner–Carleman type, via decomposition by support or growth order into proper convex cones. They form a nuclear Fréchet space, dual to the nuclear (DF)-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420176.png" /> of functions holomorphic and exponentially decreasing on a strip neighbourhood of the real axis. The latter is invariant under the classical [[Fourier transform|Fourier transform]] and the [[Parseval equality|Parseval equality]] holds with this duality.

A real-analytic coordinate transformation of hyperfunctions is defined naturally via transformation of defining functions. Hence hyperfunctions can be defined on real-analytic manifolds. Fourier series are typical examples of hyperfunctions on a manifold: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420177.png" /> converges as a hyperfunction if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420178.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420179.png" />. Resolution by differential forms with hyperfunction coefficients supplies a concrete flabby resolution of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420180.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048420/h048420181.png" />, and serves to calculate relative cohomology groups.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Sato,   "Theory of hyperfunctions"  ''SÛgaku'' , '''10'''  (1958)  pp. 1–27  ((in Japanese))</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Sato,   "Theory of hyperfunctions I, II"  ''J. Fac. Sci. Univ. Tokyo Sect. 1'' , '''8'''  (1959–1960)  pp. 139–193; 387–437</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Martineau,   "Les hyperfonctions de M. Sato" , ''Sem. Bourbaki''  (1960)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Harvey,   "Hyperfunctions and linear partial differential equations" , Stanford Univ. Press  (1966)  (Thesis)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Komatsu,   "Sato's hyperfunctions and linear partial differential equations with constant coefficients"  ''Seminar Notes Univ. of Tokyo'' , '''22'''  (1970)  ((in Japanese))</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Schapira,   "Théorie des hyperfonctions" , ''Lect. notes in math.'' , '''126''' , Springer  (1970)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Morimoto,   "Introduction to Sato hyperfunctions" , Kyôritsu  (1976)  ((in Japanese))</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Kashiwara,   T. Kawai,   T. Kimura,   "Foundation of algebraic analysis" , Princeton Univ. Press  (1986)  ((translated from the Japanese))</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A. Kaneko,   "Introduction to hyperfunctions" , Kluwer  (1988)  ((translated from the Japanese))</TD></TR></table>