Some Tauberian conditions for Cesàro summability method

Rok, strany: 2012, 271 - 280

Let $u=(u_n)$ be a sequence of real numbers whose generator sequence is Cesàro summable to a finite number. We prove that $(u_n)$ is slowly oscillating if the sequence of Cesàro means of $(ω _n^(m-1)(u))$ is increasing and the following two conditions are hold:

$$ (λ -1)\limsup_n (((1) / ([λ n]-n)) ∑_k=n+1^{[λ n]} (ω _k^(m)(u))^q)^{((1) / (q))}=o(1), λ \to 1⁺, q>1, (1-λ)\limsup_n (((1) / (n-[λ n])) ∑_{k=[λ n]+1}ⁿ (ω _k^(m)(u))^q)^{((1) / (q))}=o(1), λ \to 1^-, q>1, $$

where $(ω _n^(m)(u))$ is the general control modulo of the oscillatory behavior of integer order $m≥ 1$ of a sequence $(u_n)$ defined in [D\.{I}K, F.: \textit{Tauberian theorems for convergence and subsequential convergence with moderately oscillatory behavior}, Math. Morav. \textbf{5}, (2001), 19–56] and $[λ n]$ denotes the integer part of $λ n$.

ISO 690:

Çanak, I., Totur, Ü. 2012. Some Tauberian conditions for Cesàro summability method. In Mathematica Slovaca, vol. 62, no.2, pp. 271-280. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0008-y

APA:

Çanak, I., Totur, Ü. (2012). Some Tauberian conditions for Cesàro summability method. Mathematica Slovaca, 62(2), 271-280. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0008-y