Abelian and Tauberian theorems for integrals

A. F. Grishin; I. V. Poedintseva

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arXiv · 1212.5686

Abelian and Tauberian theorems for integrals

A. F. Grishin·I. V. Poedintseva

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Abstract

A new method of obtaining Abelian and Tauberian theorems for the integral of the form $\int\limits_0^\infty K(\frac{t}{r}) dμ(t)$ is proposed. It is based on the use of limit sets of the measures. A version of Azarin's sets is constructed for Radon's measures on the ray $(0,\infty)$. Abelian theorems of a new type are proved in which asymptotic behavior of the integral is described in terms of these limit sets. Using these theorems together with an improved version of the well-known Carleman's theorem on analytic continuation, a substantial improvement of the second Wiener Tauberian theorem is obtained. Reference: 25 units. Keywords: proximate order of Valiron, Radon's measures, Azarin's limit set of measure, Azarin's regular measure, Tauberian theorem of Wiener.

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A. F. Grishin, I. V. Poedintseva. 2013-01-23. Abelian and Tauberian theorems for integrals. https://arxiv.org/abs/1212.5686

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