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Izv. RAN. Ser. Mat., 2000, Volume 64, Issue 1, Pages 37–94 (Mi izv274)  

This article is cited in 5 scientific papers (total in 5 papers)

Tauberian theorem for generalized multiplicative convolutions

Yu. N. Drozhzhinov, B. I. Zavialov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The following problem is discussed. Let $f$ be a generalized function of slow growth with support on the positive semi-axis, and let $\varphi_k$ be a sequence of “test” functions such that $\varphi_k\to\varphi_0$ as $k\to+\infty$ in some function space. Assume that the following limit exists: $\frac1{\rho(k)}(f(kt),\varphi_k(t))\to c$ where $\rho(k)$ is a regularly varying function. Find conditions under which the limit $\frac1{\rho(k)}(f(kt),\varphi(t))\to c_\varphi$, $k\to+\infty$, exists for all test functions $\varphi$. We state and prove theorems that solve this problem and apply them to the problem of existence of quasi-asymptotics for the solution of an ordinary differential equation with variable coefficients. We prove Abelian and Tauberian theorems for a wide class of integral transformations of distributions, for example, the generalized Stieltjes integral transformation.

DOI: https://doi.org/10.4213/im274

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English version:
Izvestiya: Mathematics, 2000, 64:1, 35–92

Bibliographic databases:

Document Type: Article
MSC: 40E05, 32A40, 46F12
Received: 24.06.1999

Citation: Yu. N. Drozhzhinov, B. I. Zavialov, “Tauberian theorem for generalized multiplicative convolutions”, Izv. RAN. Ser. Mat., 64:1 (2000), 37–94; Izv. Math., 64:1 (2000), 35–92

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Yu. N. Drozhzhinov, B. I. Zavialov, “Tauberian theorems for generalized functions with values in Banach spaces”, Izv. Math., 66:4 (2002), 701–769  mathnet  crossref  crossref  mathscinet  zmath  elib
    2. Yu. N. Drozhzhinov, B. I. Zavialov, “Multidimensional Tauberian theorems for Banach-space valued generalized functions”, Sb. Math., 194:11 (2003), 1599–1646  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Jasson Vindas, Stevan Pilipović, Dušan Rakić, “Tauberian Theorems for the Wavelet Transform”, J Fourier Anal Appl, 2010  crossref  mathscinet  isi  scopus
    4. Pilipovic S. Vindas J., “Multidimensional Tauberian Theorems For Vector-Valued Distributions”, Publ. Inst. Math.-Beograd, 95:109 (2014), 1–28  crossref  mathscinet  zmath  isi  scopus
    5. Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions”, Russian Math. Surveys, 71:6 (2016), 1081–1134  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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