Fourier-Borel transform

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

One says that a compact set is a carrier for an analytic functional if for every open neighbourhood of there exists a positive constant such that, for every ,

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

Let . The Fourier–Borel transform is defined by

where

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

It is immediate to show that is an entire function. Moreover, since the exponentials are dense in , 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 is carried by a compact convex set , then for every there exists a number such that, for any ,

where is the support function of .

A fundamental result in the theory of the Fourier–Borel transform is the fact that the converse is true as well: Let be an entire function. Suppose that for some compact convex set and for every there exists a number such that, for any ,

(a1)

Then is the Fourier–Borel transform of an analytic functional carried by .

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

In particular, the Fourier–Borel transform establishes an isomorphism between the space and the space of entire functions of exponential type, i.e. those entire functions for which there are positive constants , such that

If is endowed with the strong topology, and 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 . In this case, a function which satisfies the estimate (a1), i.e.

is said to be of exponential type zero, or of infra-exponential type. Given such a function , there exists a unique analytic functional such that ; such a functional is carried by and therefore is a continuous linear functional on any space , for an open subset of containing the origin. If one denotes by the space of germs of holomorphic functions at the origin (cf. also Germ), then , 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 . As an example, consider the problem of surjectivity. Let be an open convex subset of and let be carried by a compact set . Then the convolution operator

is defined by

One can show (see [a5] or [a1] and the references therein) that if is of completely regular growth and the radial regularized indicatrix of coincides with , then is a surjective operator. The converse is true provided that is bounded, strictly convex, with 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