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Year 2021, Volume: 3 Issue: 1, 15 - 19, 29.04.2021
https://doi.org/10.47087/mjm.896657

Abstract

References

  • \'{A}. Fekete, and F. M\'{o}ricz, Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over $R_{+}$. II, Publ. Math. Debrecen. 67 (1-2) (2005) 65-78.
  • G. H. Hardy, Theorems relating to the summability and convergence of slowly oscillating series, Proc. London Math. Soc.(2). 8 (1910) 310-320.
  • \c{C}. Kambak, and \.{I}. \c{C}anak, An alternative proof of a Tauberian theorem for the weighted mean summability of integrals over $R_+$, Creat. Math. Inform. 29 (1) (2020) 45-50.
  • J. Karamata, Sur les th\'{e}or\`{e}ms inverses de proc\'{e}d\'{e}s de sommabilit\'{e}, Hermann et Cie, Paris, 1937.
  • F. M\'{o}ricz, Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over $R_+$, Math. Inequal. Appl. 7 (1) (2004) 87-93.
  • A. Peyerimhoff, Lectures on summability, Springer, Berlin, 1969.
  • R. Schmidt, \"{U}ber divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1925) 89-152.

On a mean method of summability

Year 2021, Volume: 3 Issue: 1, 15 - 19, 29.04.2021
https://doi.org/10.47087/mjm.896657

Abstract

Let $p(x)$ be a nondecreasing real-valued continuous function
on $R_+:=[0,\infty)$ such that $p(0)=0$ and $p(x) \to \infty$ as $x \to \infty$.
Given a real or complex-valued integrable function $f$ in Lebesgue's sense on every bounded interval $(0,x)$
for $x>0$, in symbol $f \in L^1_{loc} (R_+)$, we set
$$
s(x)=\int _{0}^{x}f(u)du
$$
and
$$
\sigma _{p}(s(x))=\frac{1}{p(x)}\int_{0}^{x}s(u)dp(u),\,\,\,\,x>0
$$
provided that $p(x)>0$.

A function $s(x)$
is said to be summable to $l$ by the weighted mean method determined
by the function $p(x)$, in short, $(\overline{N},p)$ summable to $l$,
if
$$
\lim_{x \to \infty}\sigma _{p}(s(x))=l.
$$

If the limit $\lim _{x \to \infty} s(x)=l$
exists, then $\lim _{x \to \infty} \sigma _{p}(s(x))=l$ also exists. However, the converse is not true in general.
In this paper, we give an alternative proof a Tauberian theorem stating that convergence follows from summability by weighted mean method on $R_+:=[0,\infty)$ and a Tauberian condition of slowly decreasing type with respect to the weight function due to Karamata. These Tauberian conditions are one-sided or two-sided if $f(x)$ is a real or complex-valued function, respectively. Alternative proofs of some well-known Tauberian theorems given for several important summability methods can be obtained by choosing some particular weight functions.

References

  • \'{A}. Fekete, and F. M\'{o}ricz, Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over $R_{+}$. II, Publ. Math. Debrecen. 67 (1-2) (2005) 65-78.
  • G. H. Hardy, Theorems relating to the summability and convergence of slowly oscillating series, Proc. London Math. Soc.(2). 8 (1910) 310-320.
  • \c{C}. Kambak, and \.{I}. \c{C}anak, An alternative proof of a Tauberian theorem for the weighted mean summability of integrals over $R_+$, Creat. Math. Inform. 29 (1) (2020) 45-50.
  • J. Karamata, Sur les th\'{e}or\`{e}ms inverses de proc\'{e}d\'{e}s de sommabilit\'{e}, Hermann et Cie, Paris, 1937.
  • F. M\'{o}ricz, Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over $R_+$, Math. Inequal. Appl. 7 (1) (2004) 87-93.
  • A. Peyerimhoff, Lectures on summability, Springer, Berlin, 1969.
  • R. Schmidt, \"{U}ber divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1925) 89-152.
There are 7 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

İbrahim Çanak 0000-0002-1754-1685

Publication Date April 29, 2021
Acceptance Date March 22, 2021
Published in Issue Year 2021 Volume: 3 Issue: 1

Cite

APA Çanak, İ. (2021). On a mean method of summability. Maltepe Journal of Mathematics, 3(1), 15-19. https://doi.org/10.47087/mjm.896657
AMA Çanak İ. On a mean method of summability. Maltepe Journal of Mathematics. April 2021;3(1):15-19. doi:10.47087/mjm.896657
Chicago Çanak, İbrahim. “On a Mean Method of Summability”. Maltepe Journal of Mathematics 3, no. 1 (April 2021): 15-19. https://doi.org/10.47087/mjm.896657.
EndNote Çanak İ (April 1, 2021) On a mean method of summability. Maltepe Journal of Mathematics 3 1 15–19.
IEEE İ. Çanak, “On a mean method of summability”, Maltepe Journal of Mathematics, vol. 3, no. 1, pp. 15–19, 2021, doi: 10.47087/mjm.896657.
ISNAD Çanak, İbrahim. “On a Mean Method of Summability”. Maltepe Journal of Mathematics 3/1 (April 2021), 15-19. https://doi.org/10.47087/mjm.896657.
JAMA Çanak İ. On a mean method of summability. Maltepe Journal of Mathematics. 2021;3:15–19.
MLA Çanak, İbrahim. “On a Mean Method of Summability”. Maltepe Journal of Mathematics, vol. 3, no. 1, 2021, pp. 15-19, doi:10.47087/mjm.896657.
Vancouver Çanak İ. On a mean method of summability. Maltepe Journal of Mathematics. 2021;3(1):15-9.

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