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
,
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General references for these notions are [a3], [a5].
Let . The Fourier–Borel transform
is defined by
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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.
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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 |
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