[Paper Review] Abelian and Tauberian theorems for integrals

A. F. Grishin, I. V. Poedintseva|arXiv (Cornell University)|Dec 22, 2012

Advanced Mathematical Modeling in Engineering8 references3 citations

TL;DR

This paper introduces a novel method for deriving Abelian and Tauberian theorems for integrals of the form $\int_0^\infty K(t/r)\,d\mu(t)$ using limit sets of Radon measures on $(0,\infty)$. By generalizing Azarin's theory of limit sets to Radon measures and employing an improved version of Carleman's analytic continuation lemma, the authors establish new Abelian theorems describing asymptotic behavior via these limit sets and significantly strengthen the second Wiener Tauberian theorem.

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.

Motivation & Objective

• To develop a new method for deriving Abelian and Tauberian theorems for integrals involving a kernel $K(t/r)$ and a Radon measure $\mu$ on $(0,\infty)$.
• To generalize Azarin's theory of limit sets of measures to Radon measures on the positive half-line, enabling asymptotic analysis of integrals.
• To establish new types of Abelian theorems where the asymptotic behavior of the integral is described in terms of the limit set of the measure $\mu$.
• To apply these results, particularly via an enhanced version of Carleman's analytic continuation lemma, to achieve a substantial strengthening of the second Wiener Tauberian theorem.

Proposed method

• The method is based on defining and analyzing the limit set $Fr[\mu] = Fr[\rho(r), \mu]$ of a Radon measure $\mu$ on $(0,\infty)$, where $\mu_t(E) = \mu(tE)/V(t)$ and $V(r) = r^{\rho(r)}$ with $\rho(r)$ being the precise order of the measure.
• The authors define two classes of measures: $\mathfrak{M}_\infty(\rho(r))$ and $\mathfrak{M}(\rho(r))$, based on the growth of $\mu([r, er])$ relative to $V(r)$.
• They introduce the concept of a regular measure (in Azarin's sense) as one for which $Fr[\mu]$ consists of a single measure, which must have density $ct^{\rho-1}$.
• The key technical tool is an improved version of Carleman's lemma on analytic continuation, used to derive strong Tauberian results.
• The method relies on the wide convergence of measures $\mu_{t_n}$ to elements of $Fr[\mu]$ as $t_n \to \infty$.
• The asymptotic behavior of the integral $\Psi(r) = \int_0^\infty K(t/r)\,d\mu(t)$ is analyzed via the limit set $Fr[\mu]$ and the associated function $J(r) = \Psi(r)/V(r)$.

Experimental results

Research questions

• RQ1 How can Abelian theorems for integrals $\int_0^\infty K(t/r)\,d\mu(t)$ be formulated using the limit set of the measure $\mu$?
• RQ2 What is the precise characterization of the limit set $Fr[\mu]$ for Radon measures on $(0,\infty)$, and how does it relate to the precise order $\rho(r)$?
• RQ3 Can the second Wiener Tauberian theorem be significantly strengthened using this new framework?
• RQ4 What conditions on $\mu$ ensure that the associated integral $\Psi(r)$ has a regular limit set, and how does this relate to the regularity of $\mu$?
• RQ5 How does the improved Carleman lemma contribute to the proof of stronger Tauberian results?

Key findings

• The authors prove new Abelian theorems where the asymptotic behavior of $\Psi(r) = \int_0^\infty K(t/r)\,d\mu(t)$ is described in terms of the limit set $Fr[\mu]$ of the measure $\mu$.
• For a measure $\mu \in \mathfrak{M}_\infty(\rho(r))$, the limit set $Fr[\mu]$ is non-empty and compact in the topology of wide convergence of measures.
• If the measure $s$ with $ds(t) = \Psi(t)dt$ is regular with respect to the precise order $\rho(r)+1$, then $\mu$ is regular with respect to $\rho(r)$, and $Fr[\mu]$ consists of a single measure with density $\frac{c}{c_1}t^{\rho-1}$, where $c_1 = \int_0^\infty K(t)t^{\rho-1}dt$.
• The paper establishes that the regularity of $\mu$ is equivalent to the existence of a limit $\lim_{r\to\infty} \mu([ar,br])/V(r) = c(b^\rho - a^\rho)/\rho$ for $0 < a < b < \infty$, with the case $\rho=0$ interpreted by continuity.
• The second Wiener Tauberian theorem is significantly strengthened: the new result applies to a broader class of functions and provides sharper asymptotic control via the precise order and limit set structure.
• The improved version of Carleman's lemma on analytic continuation is instrumental in proving the strengthened Tauberian result, allowing the extension of analyticity properties from the integral to the measure.