1 file | 650.08 KB

# On Tauber's second Tauberian theorem

(2012) 64(4). p.539-560
Author
Organization
Abstract
We study Tauberian conditions for the existence of Cesàro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber's second theorem on the converse of Abel's theorem. For Schwartz distributions, we obtain extensions of many classical Tauberians for Cesàro and Abel summability of functions and measures. We give general Tauberian conditions in order to guarantee $(\mathrm{C},beta)$ summability for a given order $\beta$. The results are directly applicable to series and Stieltjes integrals, and we therefore recover the classical cases and provide new Tauberians for the converse of Abel's theorem where the conclusion is Cesàro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Szàsz for Dirichlet series.
Keywords
Laplace transform, Cesàro summability, asymptotic behavior of generalized functions, boundary behavior of analytic functions, distributional point values, Szász Tauberians, Hardy-Littlewood Tauberians, Tauberian theorems, converse of Abel's theorem, DISTRIBUTIONAL POINT VALUES, DIRICHLETS SERIES, FOURIER-SERIES, POWER-SERIES, LITTLEWOOD, BEHAVIOR, CONVERSE, HARDY

## Downloads

• On Tauber second theorem.pdf
• full text
• |
• open access
• |
• PDF
• |
• 650.08 KB

## Citation

Please use this url to cite or link to this publication:

Chicago
Estrada, Ricardo, and Jasson Vindas Diaz. 2012. “On Tauber’s Second Tauberian Theorem.” Tohoku Mathematical Journal 64 (4): 539–560.
APA
Estrada, R., & Vindas Diaz, J. (2012). On Tauber’s second Tauberian theorem. TOHOKU MATHEMATICAL JOURNAL, 64(4), 539–560.
Vancouver
1.
Estrada R, Vindas Diaz J. On Tauber’s second Tauberian theorem. TOHOKU MATHEMATICAL JOURNAL. 2012;64(4):539–60.
MLA
Estrada, Ricardo, and Jasson Vindas Diaz. “On Tauber’s Second Tauberian Theorem.” TOHOKU MATHEMATICAL JOURNAL 64.4 (2012): 539–560. Print.
@article{3141012,
abstract     = {We study Tauberian conditions for the existence of Ces{\a}ro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber's second theorem on the converse of Abel's theorem. For Schwartz distributions, we obtain extensions of many classical Tauberians for Ces{\a}ro and Abel summability of functions and measures. We give general Tauberian conditions in order to guarantee \$({\textbackslash}mathrm\{C\},beta)\$ summability for a given order \${\textbackslash}beta\$. The results are directly applicable to series and Stieltjes integrals, and we therefore recover the classical cases and provide new Tauberians for the converse of Abel's theorem where the conclusion is Ces{\a}ro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Sz{\a}sz for Dirichlet series.},
author       = {Estrada, Ricardo and Vindas Diaz, Jasson},
issn         = {0040-8735},
journal      = {TOHOKU MATHEMATICAL JOURNAL},
keyword      = {Laplace transform,Ces{\a}ro summability,asymptotic behavior of generalized functions,boundary behavior of analytic functions,distributional point values,Sz{\'a}sz Tauberians,Hardy-Littlewood Tauberians,Tauberian theorems,converse of Abel's theorem,DISTRIBUTIONAL POINT VALUES,DIRICHLETS SERIES,FOURIER-SERIES,POWER-SERIES,LITTLEWOOD,BEHAVIOR,CONVERSE,HARDY},
language     = {eng},
number       = {4},
pages        = {539--560},
title        = {On Tauber's second Tauberian theorem},
url          = {http://dx.doi.org/10.2748/tmj/1356038977},
volume       = {64},
year         = {2012},
}

`
Altmetric
View in Altmetric
Web of Science
Times cited: