Borel transform

An integral transform of the type

where is an entire function of exponential type. The Borel transform is a special case of the Laplace transform. The function is called the Borel transform of . If

then

the series converges for , where is the type of . Let be the smallest closed convex set containing all the singularities of the function ; let

be the supporting function of ; and let be the growth indicator function of ; then . If in a Borel transform the integration takes place over a ray , the corresponding integral will converge in the half-plane . Let be a closed contour surrounding ; then

If additional conditions are imposed, other representations may be deduced from this formula. Thus, consider the class of entire functions of exponential type for which

This class is identical with the class of functions that can be represented as

where .

References

[1]	E. Borel, "Leçons sur les series divergentes" , Gauthier-Villars (1928) Zbl 54.0223.01
[2]	M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)

Comments

The statement at the end of the article above is called the Paley–Wiener theorem.

References

[a1]	R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201