Distributional versions of Littlewood's Tauberian theorem

Ricardo Estrada; Jasson Vindas

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 2, page 403-420
  • ISSN: 0011-4642

Abstract

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We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.

How to cite

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Estrada, Ricardo, and Vindas, Jasson. "Distributional versions of Littlewood's Tauberian theorem." Czechoslovak Mathematical Journal 63.2 (2013): 403-420. <http://eudml.org/doc/260724>.

@article{Estrada2013,
abstract = {We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.},
author = {Estrada, Ricardo, Vindas, Jasson},
journal = {Czechoslovak Mathematical Journal},
keywords = {Tauberian theorem; Laplace transform; the converse of Abel's theorem; Littlewood's Tauberian theorem; Abel and Cesàro summability; distributional Tauberian theorem; asymptotic behavior of generalized function; Tauberian theorem; Laplace transform; converse of Abel's theorem; Littlewood's Tauberian theorem; Abel summability; Cesàro summability; distributional Tauberian theorem; asymptotic behavior; generalized function},
language = {eng},
number = {2},
pages = {403-420},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Distributional versions of Littlewood's Tauberian theorem},
url = {http://eudml.org/doc/260724},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Estrada, Ricardo
AU - Vindas, Jasson
TI - Distributional versions of Littlewood's Tauberian theorem
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 403
EP - 420
AB - We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.
LA - eng
KW - Tauberian theorem; Laplace transform; the converse of Abel's theorem; Littlewood's Tauberian theorem; Abel and Cesàro summability; distributional Tauberian theorem; asymptotic behavior of generalized function; Tauberian theorem; Laplace transform; converse of Abel's theorem; Littlewood's Tauberian theorem; Abel summability; Cesàro summability; distributional Tauberian theorem; asymptotic behavior; generalized function
UR - http://eudml.org/doc/260724
ER -

References

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