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). 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).

Sato gave a justification for the one-variable case in [a1]. A foundation for the general $n$-dimensional case was given by T. Kawai [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 $\mathcal{P}_{*}$ be the fundamental space of test functions $\varphi ( x )$ that can be analytically continued to a strip $| \operatorname { Im } z | < \delta$ and that satisfy in this strip the estimate

\begin{equation} \tag{a1} | \varphi ( z ) | e ^ { \delta | z | } < \infty \text { for some } \delta > 0. \end{equation}

This is the inductive limit of the space ${\cal P} _{ * } ^ { -\delta }$ defined by a fixed $\delta > 0$ as above, endowed with the norm

\begin{equation*} \| \varphi \| = \operatorname { sup } _ { | \operatorname {Im } z|< \delta } | \varphi ( z ) | e ^ { \delta | \operatorname { Re } z | }. \end{equation*}

The same limit space is obtained if ${\cal P} _{ * } ^ { -\delta }$ is replaced by the Hilbert space of holomorphic functions such that $| \varphi ( z ) | ^ { 2 } e ^ { \delta | z | }$ is integrable on $| \operatorname { Im } z | < \delta$. Hence, $\mathcal{P}_{*}$ becomes a well-behaved space of type (DFS) (cf. also Generalized functions, space of), even nuclear (cf. also Nuclear space), and one can consider the dual space $\mathcal{Q}$ of $\mathcal{P}_{*}$, which is called the space of Fourier hyperfunctions. $\mathcal{Q}$ is of type FS and nuclear. The classical Fourier transform is easily seen to not only preserve the space $\mathcal{P}_{*}$, but also to act on it as a topological isomorphism. Thus, one can define the Fourier transform on $\mathcal{Q}$ by duality; it also gives an isomorphism. Since there is a continuous injection with dense range $\mathcal{P} _{*} \hookrightarrow \mathcal{S}$, the same is true for $\cal S ^ { \prime } \hookrightarrow Q$. 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), a Fourier hyperfunction admits the following representation:

\begin{equation} \tag{a2} f ( x ) = \sum _ { j = 1 } ^ { N } F _ { j } ( x + i \Gamma _ { j } 0 ). \end{equation}

Here, each $\Gamma j$ is a convex open cone with vertex at the origin and $F _ { j } ( z )$ is holomorphic on the wedge $\mathbf{R} ^ { n } + i \Gamma _ { j }$ satisfying the infra-exponential estimate "for all e>0, Fjz= OeeRez" locally uniformly in $\operatorname { Im } z \in \Gamma _ { j }$, where $F _ { j } ( x + i \Gamma _ { j } 0 )$ denotes its abstract limit to the real axis. The duality with $\varphi \in \mathcal{P}_{*}$ is represented by the integral

\begin{equation*} \langle f , \varphi \rangle = \sum _ { j = 1 } ^ { N } \int _ { \gamma _ { j } } F _ { j } ( z ) \varphi ( z ) d z, \end{equation*}

where $\gamma_j$ is a path in the intersection of $\mathbf{R} ^ { n } + i \Gamma _ { j }$ with the domain of definition of $\varphi$. The value of the integral does not depend on the choice of $\gamma_j$. For the validity of all these it suffices that each $F _ { j } ( z )$ is defined only on the part of the corresponding wedge lying in a strip neighbourhood $| \operatorname { Im } z | < \delta$ of the real axis where $\varphi$ is defined.

The kernel function $e ^ { - i x s }$ of the Fourier transform is not a test function itself, but if $\operatorname { Im } \zeta$ is restricted to some convex open cone $\Delta \subset \mathbf{R} ^ { n }$, then $e ^ { - i x \zeta }$ is exponentially decreasing in $\operatorname{Re} z$ on $- \Delta ^ { \circ }$, where

\begin{equation*} \Delta ^ { \circ } = \{ x : \langle x , \eta \rangle \geq 0 \text { for all } \eta \in \Delta \} \end{equation*}

denotes the dual cone of $\Delta$. Thus, if each $F _ { j } ( z )$ is exponentially decreasing when $x \notin - \Delta ^ { \circ }$, then $F _ { j } ( z ) e ^ { - i z \zeta }$ is exponentially decreasing everywhere in $\operatorname{Re} z$ when $\operatorname { Im } z$ is in $\Gamma j$ and small enough. Thus, the Fourier transform can be calculated as the abstract limit $G ( \xi + i \Delta 0 )$ of the function

\begin{equation} \tag{a3} G ( \zeta ) = \sum _ { j = 1 } ^ { N } \int _ { \gamma _ { j } } F _ { j } ( z ) e ^ { - i z \zeta } d z. \end{equation}

For the general case one uses a partition of unity $\{ \chi _ { k } ( z ) \}$ such that each $\chi _ { k } ( z )$ is exponentially decreasing when $\operatorname{Re} z$ is outside a convex cone $- \Delta _ { k } ^ { 0 }$, and one sets

\begin{equation*} g ( \xi ) = {\cal F} [ f ] = \sum _ { k = 1 } ^ { M } G _ { k } ( \xi + i \Delta _ { k } 0 ), \end{equation*}

where each $G _ { k } ( \zeta )$ is calculated by (a3) with $\Delta$ replaced by $\Delta _ { k }$ and $F _ { j } ( z )$ by $F _ { j } ( z ) \chi _ { k } ( z )$. If the partition is made of orthants $\Delta _ { \sigma } = \{ x \in \mathbf{R} ^ { n } : \sigma _ { j }\, x _ { j } > 0 \}$ with $\sigma _ { j } = \pm 1$, then one can take as the partition function $\chi _ { \sigma } = \prod _ { j = 1 } ^ { n } 1 / ( e ^ { \sigma _ { j } z _ { j } } + 1 )$.

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 $\mathcal{P}_{*}$, one obtain a wider extension thereof. The most significant feature of Fourier hyperfunctions among such is localizability. Namely, one can define a sheaf $\mathcal{Q}$ on the directional compactification $D ^ { n } = \mathbf{R} ^ { n } \cup S _ { \infty } ^ { n - 1 }$ such that the above-introduced space of Fourier hyperfunctions agrees with the global section space $\mathcal{Q} ( D ^ { n } )$. In this sense, infra-exponential growth is the best possible choice. The sheaf $\mathcal{Q}$ of Fourier hyperfunctions is constructed from the sheaf $\tilde{\mathcal{O}}$ of germs of holomorphic functions with infra-exponential growth in the real direction as its $n$th derived sheaf: $\mathcal{Q} = \mathcal{H} _ { D ^ { n } } ( \tilde { \mathcal{O} } )$. The sheaf $\tilde{\mathcal{O}}$ is considered as living on $D ^ { n } + i {\bf R} ^ { n }$, the growth condition describing the stalks at the points at infinity. Thus, when restricted to the finite points, $\tilde{\mathcal{O}}$ reduces to $\mathcal{O}$ and $\mathcal{Q}$ to the sheaf $\mathcal{B}$ of usual hyperfunctions. Thanks to fundamental cohomology vanishing theorems for the sheaf $\tilde{\mathcal{O}}$ similar to those for $\mathcal{O}$, the space of Fourier hyperfunctions on an open set $\Omega \subset D ^ { n }$ can be represented by the global cohomology group $H _ { \Omega } ^ { n } ( U , \widetilde { \mathcal O } )$, and this in turn can be represented by the covering cohomology: Choosing $U$ to be $\tilde{\mathcal{O}}$-Stein, i.e. cohomologically trivial for $\tilde{\mathcal{O}}$ (cf. also Stein manifold), one obtains, e.g.,

\begin{equation*} {\cal Q} ( \Omega ) = \tilde {\cal O } ( U \# \Omega ) / \sum _ { j = 1 } ^ { n } \tilde {\cal O } ( U \#_j \Omega ), \end{equation*}

where

\begin{equation*} U \# \Omega = U \bigcap \{ \operatorname { Im } z _ { k } \neq 0 : k = 1 , \ldots , n \}, \end{equation*}

\begin{equation*} U \# _j \Omega = U \bigcap \{ \operatorname { Im } z _ { k } \neq 0 : k \neq j \}. \end{equation*}

This can be interpreted as the local boundary value representation

\begin{equation*} f ( x ) = \sum _ { \sigma } F _ { \sigma } ( x + i \Gamma _ { \sigma } 0 ), \end{equation*}

where

\begin{equation*} F _ { \sigma } \in \widetilde { \mathcal{O} } ( ( \Omega + \Gamma _ { \sigma } ) \cap U ). \end{equation*}

A more sophisticated choice of an $\tilde{\mathcal{O}}$-Stein covering of $U \backslash \Omega$ justifies a local boundary value representation of the form (a2) which is valid on $\Omega$, just as in the case of ordinary hyperfunctions.

The sheaf $\mathcal{Q}$ can be constructed also via duality, as in Martineau's theory for ordinary hyperfunctions: For each compact subset $K \subset D ^ { n }$, one denotes by $\mathcal{P}_{*} ( K )$ the space of $\varphi ( x )$ such that there is a neighbourhood $U$ of $K$ in $D ^ { n } + i {\bf R} ^ { n }$ such that $\varphi$ is holomorphic in $U \cap {\bf C} ^ { n }$ and is exponentially decreasing at infinity.

Notice that the decay condition is meaningful only at points at infinity of $K$. Then its dual $\mathcal{P}_{ *} ( K ) ^ { \prime }$ gives the space of Fourier hyperfunctions ${\cal Q} [ K ]$ supported by $K$. General sections of Fourier hyperfunctions can be represented as obvious equivalence classes of locally finite sums of these. ${\cal Q} [ K ]$ can be expressed by the relative cohomology group $H _ { K } ^ { n } ( D ^ { n } + i {\bf R} ^ { n } , \tilde {\cal O } )$. This is an extension of Martineau–Harvey duality in the theory of ordinary hyperfunctions. The case $K = D ^ { n }$ 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 $\mathcal{S} ^ { \prime }$ on $D ^ { n }$ of tempered distributions defined via duality in the same way as above. The notion of localization of $\mathcal{S} ^ { \prime }$ with respect to the directional coordinates is useful. The global sections of $\mathcal{S} ^ { \prime }$ on $D ^ { n }$ give the usual space of tempered distributions, whereas its global sections on ${\bf R} ^ { n }$ lead to the usual space $\mathcal{D} ^ { \prime } ( \mathbf{R} ^ { n } )$ of distributions (cf. also Nuclear space). In this case the importance of the compactification $D ^ { n }$ is not clear, because $\mathcal{S} ^ { \prime } ( D ^ { n } ) \subset \mathcal{D} ^ { \prime } ( \mathbf{R} ^ { n } )$, hence information on ${\bf R} ^ { n }$ 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 $\mathcal{Q} ( D ^ { n } ) \rightarrow \mathcal{B} ( \mathbf{R} ^ { n } )$. The surjectivity is a part of the flabbiness of the sheaf $\mathcal{Q}$, which can be established analogously to the case of $\mathcal{B}$ (cf. also 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 $+ \infty$ in one variable is given as the boundary value $f ( x ) = F ( x + i 0 ) - F ( x - i 0 )$ of the function defined by the integral

\begin{equation*} F ( z ) = - \frac { 1 } { 2 \pi i } \int_\gamma \frac { \operatorname { exp } e ^ { \zeta ^ { 2 } } } { \zeta - z } d \zeta \end{equation*}

where $\gamma$ is a simple path starting and ending at $+ \infty$ and passing through the region where $\operatorname { exp } e ^ { \zeta ^ { 2 } }$ is decreasing, say $\operatorname { Im } \zeta ^ { 2 } = \pm \pi$.

As a consequence of localizability, one can consider the support of Fourier hyperfunctions. Furthermore, by the flabbiness of the sheaf $\mathcal{Q}$, one can decompose the support of a given Fourier hyperfunction according to any covering by closed subsets of $D ^ { n }$. In particular, given a decomposition of $D ^ { n }$ by closed convex cones $- \Delta _ { k } ^ { 0 }$ with vertex at the origin, or, more generally, by closed subsets which are asymptotically such, one can decompose $f = \sum _ { k } f _ { \Delta _ { k } }$ accordingly, in such a way that $\operatorname { supp } f _ { \Delta _ { k } } \subset - \Delta _ { k } ^ { \circ }$. Then the Fourier transform of each can be calculated as the inner product

\begin{equation*} \left( f _ { \Delta _ { k } } , e ^ { - i x \zeta } \right), \end{equation*}

which is meaningful for $\operatorname { Im } \zeta \in \Delta _ { k }$, where $e ^ { - i x \zeta }$ can serve as a test function in $x$. Thus, the Fourier transform of $f$ 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 $f _ { \Delta _ { k } }$. For example, the Fourier transform of the Poisson distribution $\sum _ { k = - \infty } ^ { \infty } \delta ( x - k )$ can be calculated by means of its natural defining function $1 / ( 1 - e ^ { 2 \pi i z } )$ by a suitable choice of the integral path corresponding to the decomposition of the support to $\overline { \mathbf{R} ^ { \pm }}$, giving $2 \pi \sum _ { k = - \infty } ^ { \infty } \delta ( \xi - 2 \pi k )$.

The following generalization of Paley–Wiener type holds: Let $K$ be a convex compact subset of $D ^ { n }$. (Here, "convex" means that $K \cap {\bf R} ^ { n }$ is convex in the usual sense and that $K \cap S _ { \infty } ^ { n - 1 }$ generates a convex cone $\Gamma ^ { \circ }$, called the asymptotic cone of $K$.) Then $\operatorname{supp} f \subset K$ if and only if there exists a $G ( \zeta ) \in \widetilde { \mathcal{O} } ( D ^ { n } - i \Gamma )$ such that for any $\varepsilon > 0$ and $\Delta \subset \subset \Gamma$, $G ( \zeta ) = O ( e ^ { \varepsilon | \zeta | + H _ { K } ( \operatorname { lm } \zeta ) } )$ uniformly on $\mathbf{R} ^ { n } - i \Delta \cap \{ | \eta | \geq \varepsilon \}$, and such that $[ \mathcal{F} f ] ( \xi ) = G ( \xi - i \Gamma 0 )$.

Microlocalization.

Just as in the case of ordinary hyperfunctions, one can consider microlocal regularity for Fourier hyperfunctions: $f ( x )$ is said to be micro-analytic at $( x _ { 0 } , \xi _ { 0 } )$ if it admits a local boundary value representation (a2) that is valid in a neighbourhood of $x _ { 0 }$ such that the half-space $\xi _ { 0 } x < 0$ meets all of $\Gamma j$. This is equivalent to saying that in a neighbourhood of $x _ { 0 }$, $f ( x )$ can be written as the sum $g + h$, where $h$ comes from a local section of $\tilde{\mathcal{O}}$ and $g$ is a global Fourier hyperfunction whose Fourier transform is zero (exponentially decreasing) on a conic neighbourhood of the direction $\xi_ { 0 }$. The set of points where $f$ is not micro-analytic is called the singular spectrum or the analytic wavefront set of $f$, and is denoted by $\text{SS} \ f$ or $\operatorname{WFA} f$. This notion includes not only that of directional analyticity, but also the directional growth property of $f$. For example, if $f$ is analytic on a strip neighbourhood of the real axis but not of infra-exponential growth, then $\text{SS} \ f$ may contain $S _ { \infty } ^ { n - 1 } \times S ^ { n - 1 }$.

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 [a6].

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

Relation to other (generalized) functions.

In addition to the space of tempered distributions $\mathcal{S} ^ { \prime }$, the space of Fourier hyperfunctions contains the space of ultra-distributions of Gevrey index $s$ and of growth order $e ^ { h |x | ^ { 1 / s } }$ 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 $D ^ { n }$, 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 $J ( D )$, that is, an infinite-order differential operator whose symbol is an entire function of order $1$ and of minimal type.

Extensions.

Modified Fourier hyperfunctions.

There are many choices for the compactification of ${\bf R} ^ { n }$ or $\mathbf{C} ^ { n }$, and one can consider corresponding versions of Fourier hyperfunctions. The most important one is defined on the real axis in the full directional compactification $D ^ { 2 n }$ of $\mathbf{C} ^ { n }$, and is called the space of modified Fourier hyperfunctions. While the typical shape of a complex fundamental neighbourhood of a real point at infinity $( \infty , 0 , \ldots , 0 )$ in the space $D ^ { n } + i {\bf R} ^ { n }$ of standard Fourier hyperfunctions has the form

\begin{equation*} \left\{ z = x + i y : x _ { 1 } > \frac { | x ^ { \prime } | + 1 } { \varepsilon } , | y | < \varepsilon \right\}, \end{equation*}

the shape in $D ^ { 2 n }$ for a modified Fourier hyperfunction is

\begin{equation*} \left\{ z = x + i y : x _ { 1 } > \frac { | x ^ { \prime } | + | y | + 1 } { \varepsilon } \right\}. \end{equation*}

The sheaf $\overset{\thickapprox} { \mathcal{O} }$ of holomorphic functions of infra-exponential growth for this modified topology is defined in an obvious manner. The sheaf $\tilde{Q}$ of modified Fourier hyperfunctions is defined from the former by the same procedure as in the standard situation. The space $\tilde { \mathcal { Q } } = \tilde { \mathcal { Q } } ( D ^ { n } )$ of global modified Fourier hyperfunctions is the dual of the space $\underline{\mathcal{O}} \approx$ of exponentially decreasing holomorphic functions defined on a "conical" complex neighbourhood $C _ { \delta } = \{ z : | \operatorname { Im } z | < \delta ( | \operatorname { Re } { z | } + 1 ) \}$ of the real axis. This modified version can be used to distinguish the analytic singular support of (Fourier) hyperfunctions: Let $K \subset D ^ { n }$ be a convex compact subset with asymptotic cone $\Gamma ^ { \circ }$. Then $f ( x ) \in \tilde { \mathcal{Q} } ( D ^ { n } )$ is a section of $\underline{\mathcal{O}} \approx$ outside $K$ if and only if there is a representation $[ \mathcal{F} f ] ( \xi ) = G ( \xi - i \Gamma 0 )$ by $G ( \zeta )$ as follows: For any $\Delta \subset \subset \Gamma$ and for any $\varepsilon > 0$, one can find a $\delta > 0$ such that $G ( \zeta )$ is holomorphic in $( \mathbf{R} ^ { n } - i \Delta ) \cap C _ { \delta }$ and $G ( \zeta ) e ^ { - \varepsilon | \operatorname { lm } \zeta | - H _ { K } ( \operatorname { lm } \zeta ) }$ is of infra-exponential growth in $| \zeta |$ locally uniformly as $| \operatorname { Im } \zeta | / | \operatorname { Re } \zeta | \rightarrow 0$. This generalizes a similar result of L. Ehrenpreis for usual $C ^ { \infty }$ 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 [a2]). Its foundation was developed in [a4] in detail. Further generalizations have made by several people (see e.g. [a8], [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 $K$, and has decay of $O ( e ^ { - \varepsilon | \operatorname { Re } z | - H _ { L } ( \operatorname { Re } z )} )$ for some $\varepsilon > 0$, where $K$ and $L$ are two convex compact sets. The Fourier transform maps this space isomorphically onto a similar space, with $K$ and $L$ replaced by $L$ and $- K$. 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 [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 $M$ one can introduce the sheaf of Fourier hyperfunctions, extending the usual sheaf of hyperfunctions, whose base is the compactification of $M$. The suitable growth condition, which is not necessarily infra-exponential, is determined from the boundary geometry of $M$. This is effectively used to study the spectral properties of elliptic operators or the boundary behaviour of the manifold itself [a10], [a11].

Analogues for other types of integral transforms.

Similar ideas can be employed to generalize the Mellin transform, the Radon transform and other integral transforms (see, e.g., [a12], [a13], [a14], [a15]).

References

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