View source for Fourier hyperfunction

The theory of Fourier hyperfunctions is a variant or generalization of the theory of Fourier transforms to wider classes of (generalized) functions than the usual ones (cf. also [[Fourier transform|Fourier transform]]). The keyword of this theory is infra-exponential growth, that is, growth less than any exponential type. Fourier transforms of functions of infra-exponential growth were considered by L. Carleman. A general theory of Fourier hyperfunctions was proposed by M. Sato at the same time as his theory of hyperfunctions, in which the transformed objects can be interpreted naturally (cf. also [[Hyperfunction|Hyperfunction]]).

Sato gave a justification for the one-variable case in [[#References|[a1]]]. A foundation for the general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f1201101.png" />-dimensional case was given by T. Kawai [[#References|[a2]]], with an application to the theory of linear partial differential equations with constant coefficients. Since then, various extensions of the theory have been proposed.

As is usual in Fourier theory, generalization can proceed along three lines: as the dual, as the generalized limit, or as the derivative of the classical Fourier transform.

==Duality theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f1201102.png" /> be the fundamental space of test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f1201103.png" /> that can be analytically continued to a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f1201104.png" /> and that satisfy in this strip the estimate

<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/f/f120/f120110/f1201105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

This is the inductive limit of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f1201106.png" /> defined by a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f1201107.png" /> as above, endowed with the norm

<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/f/f120/f120110/f1201108.png" /></td> </tr></table>

The same limit space is obtained if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f1201109.png" /> is replaced by the [[Hilbert space|Hilbert space]] of holomorphic functions such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011010.png" /> is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011011.png" />. Hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011012.png" /> becomes a well-behaved space of type (DFS) (cf. also [[Generalized functions, space of|Generalized functions, space of]]), even nuclear (cf. also [[Nuclear space|Nuclear space]]), and one can consider the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011014.png" />, which is called the space of Fourier hyperfunctions. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011015.png" /> is of type FS and nuclear. The classical Fourier transform is easily seen to not only preserve the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011016.png" />, but also to act on it as a topological isomorphism. Thus, one can define the Fourier transform on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011017.png" /> by duality; it also gives an isomorphism. Since there is a continuous injection with dense range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011018.png" />, the same is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011019.png" />. In this way a generalization of the Fourier transform is obtained that is wider than the Schwartz theory of tempered distributions.

==Boundary value representation.==
Fourier hyperfunctions can be obtained as ideal limits of holomorphic functions with a certain growth restriction. Similarly to the boundary value representation of ordinary hyperfunctions by defining holomorphic functions (cf. also [[Hyperfunction|Hyperfunction]]), a Fourier hyperfunction admits the following representation:

<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/f/f120/f120110/f12011020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Here, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011021.png" /> is a convex open cone with vertex at the origin and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011022.png" /> is holomorphic on the wedge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011023.png" /> satisfying the infra-exponential estimate "for all e&gt;0, Fjz= OeeRez" locally uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011025.png" /> denotes its abstract limit to the real axis. The duality with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011026.png" /> is represented by the integral

<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/f/f120/f120110/f12011027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011028.png" /> is a path in the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011029.png" /> with the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011030.png" />. The value of the integral does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011031.png" />. For the validity of all these it suffices that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011032.png" /> is defined only on the part of the corresponding wedge lying in a strip neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011033.png" /> of the real axis where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011034.png" /> is defined.

The kernel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011035.png" /> of the Fourier transform is not a test function itself, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011036.png" /> is restricted to some convex open cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011038.png" /> is exponentially decreasing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011040.png" />, where

<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/f/f120/f120110/f12011041.png" /></td> </tr></table>

denotes the dual cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011042.png" />. Thus, if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011043.png" /> is exponentially decreasing when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011045.png" /> is exponentially decreasing everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011046.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011047.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011048.png" /> and small enough. Thus, the Fourier transform can be calculated as the abstract limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011049.png" /> of the 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/f/f120/f120110/f12011050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

For the general case one uses a partition of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011051.png" /> such that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011052.png" /> is exponentially decreasing when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011053.png" /> is outside a convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011054.png" />, and one sets

<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/f/f120/f120110/f12011055.png" /></td> </tr></table>

where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011056.png" /> is calculated by (a3) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011057.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011059.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011060.png" />. If the partition is made of orthants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011061.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011062.png" />, then one can take as the partition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011063.png" />.

==Localization.==
There are many possibilities to extend the Fourier transform by means of duality, based on various fundamental spaces of test functions stable under the Fourier transform. If one chooses a fundamental space smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011064.png" />, one obtain a wider extension thereof. The most significant feature of Fourier hyperfunctions among such is localizability. Namely, one can define a [[Sheaf|sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011065.png" /> on the directional compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011067.png" /> such that the above-introduced space of Fourier hyperfunctions agrees with the global section space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011068.png" />. In this sense, infra-exponential growth is the best possible choice. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011069.png" /> of Fourier hyperfunctions is constructed from the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011070.png" /> of germs of holomorphic functions with infra-exponential growth in the real direction as its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011071.png" />th derived sheaf: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011072.png" />. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011073.png" /> is considered as living on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011074.png" />, the growth condition describing the stalks at the points at infinity. Thus, when restricted to the finite points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011075.png" /> reduces to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011077.png" /> to the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011078.png" /> of usual hyperfunctions. Thanks to fundamental cohomology vanishing theorems for the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011079.png" /> similar to those for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011080.png" />, the space of Fourier hyperfunctions on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011081.png" /> can be represented by the global cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011082.png" />, and this in turn can be represented by the covering cohomology: Choosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011083.png" /> to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011084.png" />-Stein, i.e. cohomologically trivial for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011085.png" /> (cf. also [[Stein manifold|Stein manifold]]), one obtains, e.g.,

<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/f/f120/f120110/f12011086.png" /></td> </tr></table>

where

<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/f/f120/f120110/f12011087.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/f/f120/f120110/f12011088.png" /></td> </tr></table>

This can be interpreted as the local boundary value representation

<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/f/f120/f120110/f12011089.png" /></td> </tr></table>

where

<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/f/f120/f120110/f12011090.png" /></td> </tr></table>

A more sophisticated choice of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011091.png" />-Stein covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011092.png" /> justifies a local boundary value representation of the form (a2) which is valid on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011093.png" />, just as in the case of ordinary hyperfunctions.

The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011094.png" /> can be constructed also via duality, as in Martineau's theory for ordinary hyperfunctions: For each compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011095.png" />, one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011096.png" /> the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011097.png" /> such that there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f12011099.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110100.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110101.png" /> is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110102.png" /> and is exponentially decreasing at infinity.

Notice that the decay condition is meaningful only at points at infinity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110103.png" />. Then its dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110104.png" /> gives the space of Fourier hyperfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110105.png" /> supported by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110106.png" />. General sections of Fourier hyperfunctions can be represented as obvious equivalence classes of locally finite sums of these. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110107.png" /> can be expressed by the relative cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110108.png" />. This is an extension of Martineau–Harvey duality in the theory of ordinary hyperfunctions. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110109.png" /> corresponds to that for global Fourier hyperfunctions, given at the beginning.

Contrary to the general feeling, Schwartz tempered distributions can be localized in a similar way: One can consider a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110110.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110111.png" /> of tempered distributions defined via duality in the same way as above. The notion of localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110112.png" /> with respect to the directional coordinates is useful. The global sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110113.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110114.png" /> give the usual space of tempered distributions, whereas its global sections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110115.png" /> lead to the usual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110116.png" /> of distributions (cf. also [[Nuclear space|Nuclear space]]). In this case the importance of the compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110117.png" /> is not clear, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110118.png" />, hence information on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110119.png" /> suffices to determine a tempered distribution, as is usually done.

In the case of Fourier hyperfunctions, however, this relation is more complicated: There is a canonical surjection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110120.png" />. The surjectivity is a part of the flabbiness of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110121.png" />, which can be established analogously to the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110122.png" /> (cf. also [[Flabby sheaf|Flabby sheaf]]). As usual, the extension is not unique. Especially, there are Fourier hyperfunctions supported at the points at infinity. The concrete Morimoto–Yoshino example of a Fourier hyperfunction with one point support at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110124.png" /> in one variable is given as the boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110125.png" /> of the function defined by the integral

<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/f/f120/f120110/f120110126.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110127.png" /> is a simple path starting and ending at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110128.png" /> and passing through the region where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110129.png" /> is decreasing, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110130.png" />.

As a consequence of localizability, one can consider the support of Fourier hyperfunctions. Furthermore, by the flabbiness of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110131.png" />, one can decompose the support of a given Fourier hyperfunction according to any covering by closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110132.png" />. In particular, given a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110133.png" /> by closed convex cones <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110134.png" /> with vertex at the origin, or, more generally, by closed subsets which are asymptotically such, one can decompose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110135.png" /> accordingly, in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110136.png" />. Then the Fourier transform of each can be calculated as the inner product

<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/f/f120/f120110/f120110137.png" /></td> </tr></table>

which is meaningful for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110138.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110139.png" /> can serve as a test function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110140.png" />. Thus, the Fourier transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110141.png" /> can be calculated as the sum of the boundary values of these from respective wedges. In practical calculations one does not have to replace the defining functions to realize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110142.png" />. For example, the Fourier transform of the Poisson distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110143.png" /> can be calculated by means of its natural defining function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110144.png" /> by a suitable choice of the integral path corresponding to the decomposition of the support to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110145.png" />, giving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110146.png" />.

The following generalization of Paley–Wiener type holds: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110147.png" /> be a convex compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110148.png" />. (Here, "convex" means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110149.png" /> is convex in the usual sense and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110150.png" /> generates a convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110151.png" />, called the asymptotic cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110152.png" />.) Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110153.png" /> if and only if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110154.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110157.png" /> uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110158.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110159.png" />.

==Microlocalization.==
Just as in the case of ordinary hyperfunctions, one can consider microlocal regularity for Fourier hyperfunctions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110160.png" /> is said to be micro-analytic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110161.png" /> if it admits a local boundary value representation (a2) that is valid in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110162.png" /> such that the half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110163.png" /> meets all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110164.png" />. This is equivalent to saying that in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110165.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110166.png" /> can be written as the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110167.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110168.png" /> comes from a local section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110170.png" /> is a global Fourier hyperfunction whose Fourier transform is zero (exponentially decreasing) on a conic neighbourhood of the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110171.png" />. The set of points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110172.png" /> is not micro-analytic is called the singular spectrum or the analytic wavefront set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110173.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110174.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110175.png" />. This notion includes not only that of directional analyticity, but also the directional growth property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110176.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110177.png" /> is analytic on a strip neighbourhood of the real axis but not of infra-exponential growth, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110178.png" /> may contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110179.png" />.

One can introduce the sheaf of Fourier microfunctions representing the microlocal singularities of the Fourier hyperfunctions. This sheaf is flabby, and consequently one can decompose the singular spectrum of Fourier hyperfunctions according to any closed covering [[#References|[a6]]].

This notion may be effectively employed for certain problems in global analysis on unbounded domains [[#References|[a5]]].

==Relation to other (generalized) functions.==
In addition to the space of tempered distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110180.png" />, the space of Fourier hyperfunctions contains the space of ultra-distributions of Gevrey index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110181.png" /> and of growth order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110182.png" /> as a subspace invariant under the Fourier transform. Hyperfunctions with compact supports can be canonically considered as Fourier hyperfunctions. General hyperfunctions can be considered as Fourier hyperfunctions after extension to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110183.png" />, but the extension is not unique and the ambiguity of extension influences the result of the Fourier transform in an essential manner. A measurable function of infra-exponential growth (in the sense of the essential supremum) can be canonically considered as a Fourier hyperfunction. Conversely, any Fourier hyperfunction can be represented as the derivative of such a function by a local operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110184.png" />, that is, an infinite-order differential operator whose symbol is an entire function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110185.png" /> and of minimal type.

==Extensions.==


===Modified Fourier hyperfunctions.===
There are many choices for the compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110186.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110187.png" />, and one can consider corresponding versions of Fourier hyperfunctions. The most important one is defined on the real axis in the full directional compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110189.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110190.png" />, and is called the space of modified Fourier hyperfunctions. While the typical shape of a complex fundamental neighbourhood of a real point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110191.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110192.png" /> of standard Fourier hyperfunctions has 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/f/f120/f120110/f120110193.png" /></td> </tr></table>

the shape in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110194.png" /> for a modified Fourier hyperfunction is

<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/f/f120/f120110/f120110195.png" /></td> </tr></table>

The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110196.png" /> of holomorphic functions of infra-exponential growth for this modified topology is defined in an obvious manner. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110197.png" /> of modified Fourier hyperfunctions is defined from the former by the same procedure as in the standard situation. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110198.png" /> of global modified Fourier hyperfunctions is the dual of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110199.png" /> of exponentially decreasing holomorphic functions defined on a "conical" complex neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110200.png" /> of the real axis. This modified version can be used to distinguish the analytic singular support of (Fourier) hyperfunctions: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110201.png" /> be a convex compact subset with asymptotic cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110202.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110203.png" /> is a section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110204.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110205.png" /> if and only if there is a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110206.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110207.png" /> as follows: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110208.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110209.png" />, one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110210.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110211.png" /> is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110212.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110213.png" /> is of infra-exponential growth in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110214.png" /> locally uniformly as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110215.png" />. This generalizes a similar result of L. Ehrenpreis for usual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110216.png" /> singular supports.

The profitability of the idea of a modified Fourier hyperfunction was discovered by M. Sato and T. Kawai in their joint researches (see [[#References|[a2]]]). Its foundation was developed in [[#References|[a4]]] in detail. Further generalizations have made by several people (see e.g. [[#References|[a8]]], [[#References|[a9]]]).

===Fourier ultra-hyperfunctions.===
Functions of exponential growth cannot be canonically considered as Fourier hyperfunctions. The theory of Fourier ultra-hyperfunctions enables one to treat them naturally: The fundamental space of test functions in this theory is defined on a neighbourhood of a convex tube of base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110217.png" />, and has decay of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110218.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110219.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110220.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110221.png" /> are two convex compact sets. The Fourier transform maps this space isomorphically onto a similar space, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110222.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110223.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110224.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110225.png" />. The elements of the dual space of this space are called Fourier ultra-hyperfunctions. They can also be given via the relative cohomology group of the corresponding sheaf of holomorphic functions with suitable growth. Thus, in short, the growth of the defining functions is allowed to be of a fixed exponential type, but as compensation for that, its "supports" as analytic functionals bulk to a tube, and no local theory is available [[#References|[a3]]]. This theory is useful for identifying special kinds of entire functions of exponential type.

===Fourier hyperfunctions on manifolds.===
On a real-analytic open manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110226.png" /> one can introduce the sheaf of Fourier hyperfunctions, extending the usual sheaf of hyperfunctions, whose base is the compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110227.png" />. The suitable growth condition, which is not necessarily infra-exponential, is determined from the boundary geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110228.png" />. This is effectively used to study the spectral properties of elliptic operators or the boundary behaviour of the manifold itself [[#References|[a10]]], [[#References|[a11]]].

===Analogues for other types of integral transforms.===
Similar ideas can be employed to generalize the [[Mellin transform|Mellin transform]], the [[Radon transform|Radon transform]] and other integral transforms (see, e.g., [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a15]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Sato, "Theory of hyperfunctions" ''SÛgaku'' , '''10''' (1958) pp. 1–27 (In Japanese) {{MR|0478241}} {{MR|0132392}} {{MR|0114124}} {{ZBL|0097.31404}} {{ZBL|0087.31402}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> T. Kawai, "On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients" ''J. Fac. Sci. Univ. Tokyo Sec. 1A'' , '''17''' (1970) pp. 467–517 {{MR|0298200}} {{ZBL|0212.46101}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Y.S. Park, M. Morimoto, "Fourier ultra hyperfunctions in the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110229.png" />-space" ''J. Fac. Sci. Univ. Tokyo Sec. 1A'' , '''20''' (1973) pp. 121–127 {{MR|324399}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> Y. Saburi, "Fundamental properties of modified Fourier hyperfunctions" ''Tokyo J. Math.'' , '''4''' (1985) pp. 231–273 {{MR|0800088}} {{ZBL|0597.46038}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A. Kaneko, "On the global existence of real analytic solutions of linear partial differential equations on unbounded domain" ''J. Fac. Sci. Univ. Tokyo Sec. 1A'' , '''32''' (1985) pp. 319–372 {{MR|0802895}} {{ZBL|0583.35013}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Kaneko, "On the flabbiness of the sheaf of Fourier hyperfunctions" ''Sci. Pap. Coll. Gen. Educ. Univ. Tokyo'' , '''36''' (1986) pp. 1–14</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A. Kaneko, "Introduction to hyperfunctions" , Kluwer Acad. Publ. (1988) {{MR|1026013}} {{ZBL|0687.46027}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> S. Nagamachi, "The theory of vector valued Fourier hyperfunctions of mixed type I" ''Publ. RIMS Kyoto Univ.'' , '''17''' (1981) pp. 25–63 {{MR|0613934}} {{ZBL|0487.46024}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> Y. Ito, "Fourier hyperfunctions of general type" ''J. Math. Kyoto Univ.'' , '''38''' (1988) pp. 213–265 {{MR|0953176}} {{ZBL|0711.46035}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Morimoto, "Analytic functionals on the Lie sphere" ''Tokyo J. Math.'' , '''3''' (1980) pp. 1–35 {{MR|0584545}} {{ZBL|0454.46032}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> T. Oshima, Y. Saburi, M. Wakayama, "Paley Wiener theorems on a symmetric space and its applications" ''Diff. Geom. Appl.'' , '''1''' (1991) pp. 247–278</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> H. Komatsu, "Multipliers for Laplace hyperfunctions: A justification of Heaviside rules" ''Proc. Steklov Inst. Math.'' , '''203''' (1994) pp. 271–279 {{MR|1382614}} {{ZBL|0897.46024}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> T. Takiguchi, A. Kaneko, "Radon transform of hyperfunctions and support theorem" ''Hokkaido Math. J.'' , '''24''' (1995) pp. 63–103 {{MR|1319031}} {{ZBL|0828.46040}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> B. Ziemian, "The Mellin transformation and multidimensional generalized Taylor expansions of singular functions" ''J. Fac. Sci. Univ. Tokyo Sec. 1A'' , '''36''' (1989) pp. 263–295 {{MR|1015000}} {{ZBL|0713.46025}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> Z. Szmydt, B. Ziemian, "Laplace distributions and hyperfunctions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120110/f120110230.png" />" ''J. Math. Sci. Univ. Tokyo'' , '''5''' (1998) pp. 41–74 {{MR|1617071}} {{ZBL|0917.46038}} </TD></TR></table>