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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200901.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200902.png" />-dimensional complex space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200903.png" /> denote the space of entire functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200904.png" /> complex variables, equipped with the topology of uniform convergence on the compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200905.png" /> (cf. also [[Entire function|Entire function]]; [[Uniform convergence|Uniform convergence]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200906.png" /> be its dual space of continuous linear functionals. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200907.png" /> are usually called analytic functionals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200908.png" />.
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One says that a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f1200909.png" /> is a carrier for an analytic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009010.png" /> if for every open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009012.png" /> there exists a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009013.png" /> such that, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009014.png" />,
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<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/f120090/f12009015.png" /></td> </tr></table>
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Let $\mathbf{C} ^ { n }$ be the $n$-dimensional complex space, and let $\mathcal{H} ( \mathbf{C} ^ { n } )$ denote the space of entire functions in $n$ complex variables, equipped with the topology of uniform convergence on the compact subsets of $\mathbf{C} ^ { n }$ (cf. also [[Entire function|Entire function]]; [[Uniform convergence|Uniform convergence]]). Let $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ be its dual space of continuous linear functionals. The elements of $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ are usually called analytic functionals in $\mathbf{C} ^ { n }$.
 +
 
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One says that a compact set $K \subseteq \mathbf{C} ^ { n }$ is a carrier for an analytic functional $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$ if for every open neighbourhood $U$ of $K$ there exists a positive constant $C _ { U }$ such that, for every $f \in \mathcal{H} ( \mathbf{C} ^ { n } )$,
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\begin{equation*} | \mu ( f ) | \leq C _ { U } \operatorname { sup } _ { U } | f ( z ) |. \end{equation*}
  
 
General references for these notions are [[#References|[a3]]], [[#References|[a5]]].
 
General references for these notions are [[#References|[a3]]], [[#References|[a5]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009016.png" />. The Fourier–Borel transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009017.png" /> is defined by
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Let $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$. The Fourier–Borel transform $\mathcal{F} \mu ( \zeta )$ is defined 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/f/f120/f120090/f12009018.png" /></td> </tr></table>
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\begin{equation*} \mathcal{F} \mu ( \zeta ) = \mu ( \operatorname { exp } \zeta z ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009019.png" />
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where $\zeta z = \zeta _ { 1 } z _ { 1 } + \ldots + \zeta _ { n } z _ { n }$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009020.png" />, the use of this transform goes back to E. Borel, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009021.png" /> it first appeared in a series of papers by A. Martineau, culminating with [[#References|[a6]]].
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For $n = 1$, the use of this transform goes back to E. Borel, while for $n &gt; 1$ it first appeared in a series of papers by A. Martineau, culminating with [[#References|[a6]]].
  
It is immediate to show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009022.png" /> is an entire function. Moreover, since the exponentials are dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009023.png" />, an analytic functional is uniquely determined by its Fourier–Borel transform.
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It is immediate to show that $\mathcal{F} \mu$ is an entire function. Moreover, since the exponentials are dense in $\mathcal{H} ( \mathbf{C} ^ { n } )$, an analytic functional is uniquely determined by its Fourier–Borel transform.
  
By using the definition of carrier of an analytic functional, it is easy to see that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009024.png" /> is carried by a compact convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009025.png" />, then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009026.png" /> there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009027.png" /> such that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009028.png" />,
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By using the definition of carrier of an analytic functional, it is easy to see that if $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$ is carried by a compact convex set $K$, then for every $\epsilon &gt; 0$ there exists a number $C _ { \epsilon } &gt; 0$ such that, for any $\zeta \in \mathbf{C} ^ { n }$,
  
<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/f120090/f12009029.png" /></td> </tr></table>
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\begin{equation*} | \mathcal{F} \mu ( \zeta ) | \leq C _ { \epsilon } \operatorname { exp } ( H _ { K } ( \zeta ) + \epsilon | \zeta | ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009030.png" /> is the support function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009031.png" />.
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where $H _ { K } ( \zeta ) = \operatorname { sup } _ { z \in K } \operatorname { Re } ( \zeta z )$ is the support function of $K$.
  
A fundamental result in the theory of the Fourier–Borel transform is the fact that the converse is true as well: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009032.png" /> be an entire function. Suppose that for some compact convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009033.png" /> and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009034.png" /> there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009035.png" /> such that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009036.png" />,
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A fundamental result in the theory of the Fourier–Borel transform is the fact that the converse is true as well: Let $f ( \zeta )$ be an entire function. Suppose that for some compact convex set $K$ and for every $\epsilon &gt; 0$ there exists a number $C _ { \epsilon } &gt; 0$ such that, for any $\zeta \in \mathbf{C} ^ { n }$,
  
<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/f120090/f12009037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} | f ( \zeta ) | \leq C _ { \epsilon } \operatorname { exp } ( H _ { K } ( \zeta ) + \epsilon | \zeta | ). \end{equation}
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009038.png" /> is the Fourier–Borel transform of an analytic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009039.png" /> carried by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009040.png" />.
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Then $f$ is the Fourier–Borel transform of an analytic functional $\mu$ carried by $K$.
  
This theorem, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009041.png" />, was proved by G. Pólya, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009042.png" /> it is due to A. Martineau [[#References|[a7]]].
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This theorem, for $n = 1$, was proved by G. Pólya, while for $n &gt; 1$ it is due to A. Martineau [[#References|[a7]]].
  
In particular, the Fourier–Borel transform establishes an isomorphism between the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009043.png" /> and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009044.png" /> of entire functions of exponential type, i.e. those entire functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009045.png" /> for which there are positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009047.png" /> such that
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In particular, the Fourier–Borel transform establishes an isomorphism between the space $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ and the space $\operatorname{Exp}( \mathbf{C} ^ { n } )$ of entire functions of exponential type, i.e. those entire functions $f$ for which there are positive constants $A$, $B$ such that
  
<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/f120090/f12009048.png" /></td> </tr></table>
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\begin{equation*} | f ( \zeta ) | \leq A\operatorname { exp } ( B | \zeta | ). \end{equation*}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009049.png" /> is endowed with the [[Strong topology|strong topology]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009050.png" /> with its natural [[Inductive limit|inductive limit]] topology, then the Fourier–Borel transform is actually a topological isomorphism, [[#References|[a2]]].
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If $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ is endowed with the [[Strong topology|strong topology]], and $\operatorname{Exp}( \mathbf{C} ^ { n } )$ with its natural [[Inductive limit|inductive limit]] topology, then the Fourier–Borel transform is actually a topological isomorphism, [[#References|[a2]]].
  
A case of particular interest occurs when, in the above assertion, one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009051.png" />. In this case, a function which satisfies the estimate (a1), i.e.
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A case of particular interest occurs when, in the above assertion, one takes $K = \{ 0 \}$. In this case, a function which satisfies the estimate (a1), i.e.
  
<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/f120090/f12009052.png" /></td> </tr></table>
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\begin{equation*} | f ( \zeta ) | \leq C _ { \epsilon } \operatorname { exp } ( \epsilon | \zeta | ) \end{equation*}
  
is said to be of exponential type zero, or of infra-exponential type. Given such a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009053.png" />, there exists a unique analytic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009054.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009055.png" />; such a functional is carried by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009056.png" /> and therefore is a continuous linear functional on any space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009057.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009058.png" /> an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009059.png" /> containing the origin. If one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009060.png" /> the space of germs of holomorphic functions at the origin (cf. also [[Germ|Germ]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009061.png" />, the space of hyperfunctions supported at the origin (cf. also [[Hyperfunction|Hyperfunction]]); the Fourier–Borel transform is therefore well defined on such a space. In fact, it is well defined on every hyperfunction with compact support. For this and related topics, see e.g. [[#References|[a1]]], [[#References|[a4]]].
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is said to be of exponential type zero, or of infra-exponential type. Given such a function $f$, there exists a unique analytic functional $\mu$ such that $\mathcal{F} \mu = f$; such a functional is carried by $K = \{ 0 \}$ and therefore is a continuous linear functional on any space $\mathcal{H} ( U )$, for $U$ an open subset of $\mathbf{C} ^ { n }$ containing the origin. If one denotes by $\mathcal{O}_{ \{ 0 \}}$ the space of germs of holomorphic functions at the origin (cf. also [[Germ|Germ]]), then $\mathcal{O} _ { \{ 0 \} } ^ { \prime } = \mathcal{B} _ { \{ 0 \} }$, the space of hyperfunctions supported at the origin (cf. also [[Hyperfunction|Hyperfunction]]); the Fourier–Borel transform is therefore well defined on such a space. In fact, it is well defined on every hyperfunction with compact support. For this and related topics, see e.g. [[#References|[a1]]], [[#References|[a4]]].
  
The Fourier–Borel transform is a central tool in the study of convolution equations in convex sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009062.png" />. As an example, consider the problem of surjectivity. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009063.png" /> be an open convex subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009064.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009065.png" /> be carried by a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009066.png" />. Then the convolution operator
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The Fourier–Borel transform is a central tool in the study of convolution equations in convex sets in $\mathbf{C} ^ { n }$. As an example, consider the problem of surjectivity. Let $\Omega$ be an open convex subset of $\mathbf{C} ^ { n }$ and let $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$ be carried by a compact set $K$. Then the convolution operator
  
<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/f120090/f12009067.png" /></td> </tr></table>
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\begin{equation*} \mu ^ { * } : {\cal H} ( \Omega + K ) \rightarrow {\cal H} ( \Omega ) \end{equation*}
  
 
is defined by
 
is defined 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/f/f120/f120090/f12009068.png" /></td> </tr></table>
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\begin{equation*} \mu ^ { * } f ( z ) = \mu ( \zeta \mapsto f ( z + \zeta ) ). \end{equation*}
  
One can show (see [[#References|[a5]]] or [[#References|[a1]]] and the references therein) that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009069.png" /> is of completely regular growth and the radial regularized indicatrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009070.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009072.png" /> is a surjective operator. The converse is true provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009073.png" /> is bounded, strictly convex, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120090/f12009074.png" /> boundary.
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One can show (see [[#References|[a5]]] or [[#References|[a1]]] and the references therein) that if $\mathcal{F} \mu$ is of completely regular growth and the radial regularized indicatrix of $\mathcal{F} \mu$ coincides with $H _ { K }$, then $\mu ^ { * }$ is a surjective operator. The converse is true provided that $\Omega$ is bounded, strictly convex, with $C ^ { 2 }$ boundary.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.A. Berenstein,  D.C. Struppa,  "Complex analysis and convolution equations" , ''Encycl. Math. Sci.'' , '''54''' , Springer  (1993)  pp. 1–108</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Ehrenpreis,  "Fourier analysis in several complex variables" , Wiley  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Kato,  D.C. Struppa,  "Fundamentals of algebraic microlocal analysis" , M. Dekker  (1999)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Lelong,  L. Gruman,  "Entire functions of several complex variables" , Springer  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Martineau,  "Sur les fonctionnelles analytiques et la transformation de Fourier–Borel"  ''J. Ann. Math. (Jerusalem)'' , '''XI'''  (1963)  pp. 1–164</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Martineau,  "Equations différentialles d'ordre infini"  ''Bull. Soc. Math. France'' , '''95'''  (1967)  pp. 109–154</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  C.A. Berenstein,  D.C. Struppa,  "Complex analysis and convolution equations" , ''Encycl. Math. Sci.'' , '''54''' , Springer  (1993)  pp. 1–108</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  L. Ehrenpreis,  "Fourier analysis in several complex variables" , Wiley  (1970)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , v. Nostrand  (1966)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G. Kato,  D.C. Struppa,  "Fundamentals of algebraic microlocal analysis" , M. Dekker  (1999)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P. Lelong,  L. Gruman,  "Entire functions of several complex variables" , Springer  (1986)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Martineau,  "Sur les fonctionnelles analytiques et la transformation de Fourier–Borel"  ''J. Ann. Math. (Jerusalem)'' , '''XI'''  (1963)  pp. 1–164</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A. Martineau,  "Equations différentialles d'ordre infini"  ''Bull. Soc. Math. France'' , '''95'''  (1967)  pp. 109–154</td></tr></table>

Revision as of 16:58, 1 July 2020

Let $\mathbf{C} ^ { n }$ be the $n$-dimensional complex space, and let $\mathcal{H} ( \mathbf{C} ^ { n } )$ denote the space of entire functions in $n$ complex variables, equipped with the topology of uniform convergence on the compact subsets of $\mathbf{C} ^ { n }$ (cf. also Entire function; Uniform convergence). Let $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ be its dual space of continuous linear functionals. The elements of $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ are usually called analytic functionals in $\mathbf{C} ^ { n }$.

One says that a compact set $K \subseteq \mathbf{C} ^ { n }$ is a carrier for an analytic functional $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$ if for every open neighbourhood $U$ of $K$ there exists a positive constant $C _ { U }$ such that, for every $f \in \mathcal{H} ( \mathbf{C} ^ { n } )$,

\begin{equation*} | \mu ( f ) | \leq C _ { U } \operatorname { sup } _ { U } | f ( z ) |. \end{equation*}

General references for these notions are [a3], [a5].

Let $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$. The Fourier–Borel transform $\mathcal{F} \mu ( \zeta )$ is defined by

\begin{equation*} \mathcal{F} \mu ( \zeta ) = \mu ( \operatorname { exp } \zeta z ), \end{equation*}

where $\zeta z = \zeta _ { 1 } z _ { 1 } + \ldots + \zeta _ { n } z _ { n }$

For $n = 1$, the use of this transform goes back to E. Borel, while for $n > 1$ it first appeared in a series of papers by A. Martineau, culminating with [a6].

It is immediate to show that $\mathcal{F} \mu$ is an entire function. Moreover, since the exponentials are dense in $\mathcal{H} ( \mathbf{C} ^ { n } )$, an analytic functional is uniquely determined by its Fourier–Borel transform.

By using the definition of carrier of an analytic functional, it is easy to see that if $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$ is carried by a compact convex set $K$, then for every $\epsilon > 0$ there exists a number $C _ { \epsilon } > 0$ such that, for any $\zeta \in \mathbf{C} ^ { n }$,

\begin{equation*} | \mathcal{F} \mu ( \zeta ) | \leq C _ { \epsilon } \operatorname { exp } ( H _ { K } ( \zeta ) + \epsilon | \zeta | ), \end{equation*}

where $H _ { K } ( \zeta ) = \operatorname { sup } _ { z \in K } \operatorname { Re } ( \zeta z )$ is the support function of $K$.

A fundamental result in the theory of the Fourier–Borel transform is the fact that the converse is true as well: Let $f ( \zeta )$ be an entire function. Suppose that for some compact convex set $K$ and for every $\epsilon > 0$ there exists a number $C _ { \epsilon } > 0$ such that, for any $\zeta \in \mathbf{C} ^ { n }$,

\begin{equation} \tag{a1} | f ( \zeta ) | \leq C _ { \epsilon } \operatorname { exp } ( H _ { K } ( \zeta ) + \epsilon | \zeta | ). \end{equation}

Then $f$ is the Fourier–Borel transform of an analytic functional $\mu$ carried by $K$.

This theorem, for $n = 1$, was proved by G. Pólya, while for $n > 1$ it is due to A. Martineau [a7].

In particular, the Fourier–Borel transform establishes an isomorphism between the space $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ and the space $\operatorname{Exp}( \mathbf{C} ^ { n } )$ of entire functions of exponential type, i.e. those entire functions $f$ for which there are positive constants $A$, $B$ such that

\begin{equation*} | f ( \zeta ) | \leq A\operatorname { exp } ( B | \zeta | ). \end{equation*}

If $\mathcal H ( \mathbf C ^ { n } ) ^ { \prime }$ is endowed with the strong topology, and $\operatorname{Exp}( \mathbf{C} ^ { n } )$ with its natural inductive limit topology, then the Fourier–Borel transform is actually a topological isomorphism, [a2].

A case of particular interest occurs when, in the above assertion, one takes $K = \{ 0 \}$. In this case, a function which satisfies the estimate (a1), i.e.

\begin{equation*} | f ( \zeta ) | \leq C _ { \epsilon } \operatorname { exp } ( \epsilon | \zeta | ) \end{equation*}

is said to be of exponential type zero, or of infra-exponential type. Given such a function $f$, there exists a unique analytic functional $\mu$ such that $\mathcal{F} \mu = f$; such a functional is carried by $K = \{ 0 \}$ and therefore is a continuous linear functional on any space $\mathcal{H} ( U )$, for $U$ an open subset of $\mathbf{C} ^ { n }$ containing the origin. If one denotes by $\mathcal{O}_{ \{ 0 \}}$ the space of germs of holomorphic functions at the origin (cf. also Germ), then $\mathcal{O} _ { \{ 0 \} } ^ { \prime } = \mathcal{B} _ { \{ 0 \} }$, the space of hyperfunctions supported at the origin (cf. also Hyperfunction); the Fourier–Borel transform is therefore well defined on such a space. In fact, it is well defined on every hyperfunction with compact support. For this and related topics, see e.g. [a1], [a4].

The Fourier–Borel transform is a central tool in the study of convolution equations in convex sets in $\mathbf{C} ^ { n }$. As an example, consider the problem of surjectivity. Let $\Omega$ be an open convex subset of $\mathbf{C} ^ { n }$ and let $\mu \in \mathcal{H} ( \mathbf{C} ^ { n } ) ^ { \prime }$ be carried by a compact set $K$. Then the convolution operator

\begin{equation*} \mu ^ { * } : {\cal H} ( \Omega + K ) \rightarrow {\cal H} ( \Omega ) \end{equation*}

is defined by

\begin{equation*} \mu ^ { * } f ( z ) = \mu ( \zeta \mapsto f ( z + \zeta ) ). \end{equation*}

One can show (see [a5] or [a1] and the references therein) that if $\mathcal{F} \mu$ is of completely regular growth and the radial regularized indicatrix of $\mathcal{F} \mu$ coincides with $H _ { K }$, then $\mu ^ { * }$ is a surjective operator. The converse is true provided that $\Omega$ is bounded, strictly convex, with $C ^ { 2 }$ boundary.

References

[a1] C.A. Berenstein, D.C. Struppa, "Complex analysis and convolution equations" , Encycl. Math. Sci. , 54 , Springer (1993) pp. 1–108
[a2] L. Ehrenpreis, "Fourier analysis in several complex variables" , Wiley (1970)
[a3] L. Hörmander, "An introduction to complex analysis in several variables" , v. Nostrand (1966)
[a4] G. Kato, D.C. Struppa, "Fundamentals of algebraic microlocal analysis" , M. Dekker (1999)
[a5] P. Lelong, L. Gruman, "Entire functions of several complex variables" , Springer (1986)
[a6] A. Martineau, "Sur les fonctionnelles analytiques et la transformation de Fourier–Borel" J. Ann. Math. (Jerusalem) , XI (1963) pp. 1–164
[a7] A. Martineau, "Equations différentialles d'ordre infini" Bull. Soc. Math. France , 95 (1967) pp. 109–154
How to Cite This Entry:
Fourier-Borel transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Borel_transform&oldid=50269
This article was adapted from an original article by D.C. Struppa (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article