Fourier-Borel transform

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, "Équations différentielles d'ordre infini" Bull. Soc. Math. France , 95 (1967) pp. 109–154