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Mat. Sb., 2016, Volume 207, Number 2, Pages 143–172 (Mi msb8507)  

This article is cited in 2 scientific papers (total in 2 papers)

A Tauberian theorem for multiple power series

A. L. Yakymiv

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Multiple sequences $\{a(i)\geqslant 0, i\in Z_+^n\}$ are considered. The notion of weak one-sided oscillation of such a sequence along a sequence
$$ \{m=m(k)=(m_1(k),…,m_n(k)), m_j(k)>0 \forall j=1,…,n, k\in \mathbb N\} $$
such that $m_j(k)\to\infty$ as $k\to\infty$ for $j=1,…,n$ is introduced. The asymptotic behaviour of the sequence $a(x_1m_1,…, x_nm_n)$ (for fixed positive numbers $x_1,…,x_n$) is deduced from the asymptotic behaviour as ${k\to\infty}$ of the generating function $A(s)$, $s\in[0,1)^n$, of the multiple sequence under consideration for $s=(e^{-\lambda_1/m_1},…,e^{-\lambda_n/m_n})$ (where $\lambda_1,…,\lambda_n$ are positive and fixed). The Tauberian theorem thus established generalizes several Tauberian theorems due to the author, which were established while investigating certain classes of random substitutions and random maps of a finite set to itself. Karamata's well-known Tauberian theorem for the generating functions of sequences was the starting point for research in this direction.
Bibliography: 36 titles.

Keywords: $\sigma$-finite measures, weak convergence of monotone functions and $\sigma$-finite measures, multiple power series, weakly one-sided oscillating multiple sequences and functions, Tauberian theorem.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00318-а
This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 14-01-00318-a).


DOI: https://doi.org/10.4213/sm8507

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English version:
Sbornik: Mathematics, 2016, 207:2, 286–313

Bibliographic databases:

Document Type: Article
UDC: 517.521.75+517.521.5
MSC: Primary 40B05, 40E05; Secondary 44A10
Received: 12.03.2015 and 04.12.2015

Citation: A. L. Yakymiv, “A Tauberian theorem for multiple power series”, Mat. Sb., 207:2 (2016), 143–172; Sb. Math., 207:2 (2016), 286–313

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions”, Russian Math. Surveys, 71:6 (2016), 1081–1134  mathnet  crossref  crossref  mathscinet  zmath  zmath  isi  elib  scopus
    2. D. V. Khlopin, “O ravnomernoi tauberovoi teoreme dlya dinamicheskikh igr”, Matem. sb., 209:1 (2018), 127–150  mathnet  crossref  mathscinet  zmath  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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