Geodesic
On inversion of the~generalized Borel transform
Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 3, pp. 817-841.

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The generalized Borel transform has a lot of applications in the theory of entire functions. It is defined on the space of functions analytic in a neighborhood of infinity and vanishing at infinity and takes values on a class $[A,+\infty)$, where $A$ is a comparison function. In this paper we obtain an integral representation of inverse generalized Borel transform for a dense class of comparison functions. This allows us to prove an analog of Polya theorem on analytic continuation of inverse Borel transform of functions of $[A,+\infty)$ for $A$ from a dense class of comparison functions of infinite order. 
@article{FPM_1999_5_3_a12,
     author = {A. Yu. Popov},
     title = {On inversion of the~generalized {Borel} transform},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {817--841},
     publisher = {mathdoc},
     volume = {5},
     number = {3},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1999_5_3_a12/}
}
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A. Yu. Popov. On inversion of the~generalized Borel transform. Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 3, pp. 817-841. http://geodesic.mathdoc.fr/item/FPM_1999_5_3_a12/