Taiwanese Journal of Mathematics

Tauberian Theorems for the Weighted Means of Measurable Functions of Several Variables

Chang-Pao Chen and Chi-Tung Chang

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Abstract

Let $f, \omega : \mathbb{R}_+^n \to \mathbb{C}$ and $T_{\omega} f(x)$ denote the weighted mean of $f$ at $x$ with respect to the weight function $\omega$. We prove that the conditions of slow oscillation and slow decrease are Tauberian conditions for the implications: $f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \rightarrow l$ and $T_{\omega} f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \rightarrow l$. We also prove that the statistical version of the conditions of deferred means are Tauberian conditions for the implication: $T_{\omega} f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \stackrel{st}{\rightarrow} l$. These generalize several well-known results.

Article information

Source
Taiwanese J. Math., Volume 15, Number 1 (2011), 181-199.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406169

Digital Object Identifier
doi:10.11650/twjm/1500406169

Mathematical Reviews number (MathSciNet)
MR2780279

Zentralblatt MATH identifier
1238.40005

Subjects
Primary: 40A30: Convergence and divergence of series and sequences of functions 40E05: Tauberian theorems, general 40G99: None of the above, but in this section

Keywords
Tauberian theorems weighted means statistical convergence

Citation

Chen, Chang-Pao; Chang, Chi-Tung. Tauberian Theorems for the Weighted Means of Measurable Functions of Several Variables. Taiwanese J. Math. 15 (2011), no. 1, 181--199. doi:10.11650/twjm/1500406169. https://projecteuclid.org/euclid.twjm/1500406169


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References

  • C.-P. Chen and C.-T. Chang, Tauberian theorems in the statistical sense for the weighted means of double sequences, Taiwanese J. Math., 11(5) (2007), 1327-1342.
  • C.-P. Chen and C.-T. Chang, Tauberian conditions under which the original convergence of double sequences follows from the statistical convergence of their weighted means, J. Math. Anal. Appl., 332(2) (2007), 1242-1248.
  • C.-P. Chen and J.-M. Hsu, Tauberian theorems for weighted means of double sequences, Anal. Math., 26 (2000), 243-262.
  • C.-P. Chen, H-C. Wu and F. Móricz, Pointwise convergence of multiple trigonometric series, J. Math. Anal. Appl., 185 (1994), 629-646.
  • Á. Fekete, Tauberian conditions under which the statistical limit of an integrable function follows from its statistical summability, Studia Sci. Math. Hungar., 43(1) (2006), 115-129.
  • Á. Fekete and F. Móricz, Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over $\mathbb{R}_+$, II, Publ. Math. Debrecen, 67(1-2) (2005), 65-78.
  • J. A. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math. Soc., 125 (1997), 3625-3631.
  • G. H. Hardy, Divergent Series, 2nd ed., Chelsea Pub. Co., New York, 1991.
  • F. Móricz, Tauberian theorems for Cesàro summable double integrals over $\mathbb{R}_+^2$, Studia Math., 138(1) (2000), 41-52.
  • F. Móricz, Tauberian theorems for double sequences that are statistically summable (C,1,1), J. Math. Anal. Appl., 286 (2003), 340-350.
  • F. Móricz, Statistical limits of measurable functions, Analysis, 24 (2004), 1-18.
  • F. Móricz, Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over $\mathbb{R}_+$, Math. Inequal. Appl., 7(1) (2004), 87-93.
  • F. Móricz, Strong Cesàro summability and statistical limit of double Fourier integrals, Acta Sci. Math. (Szeged), 71 (2005), 159-174.
  • F. Móricz, Statistical extensions of some classical Tauberian theorems in nondiscrete setting, Colloq. Math. 107(1) (2007), 45-56.
  • F. Móricz and Z. Németh, Tauberian conditions under which convergence of integrals follows from summablility $(C,1)$ over $\mathbb{R}_+$, Anal. Math., 26(1) (2000), 53-61.
  • F. Móricz and U. Stadtmüller, Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim's sence, Int. J. Math. Math. Sci., 65-68 (2004), 3499-3511.
  • W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
  • W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976.