General Tauberian theorems for the Cesàro integrability of functions

For a locally integrable function f on {[0,\infty)}, we defineF(t)=\int_{0}^{t}f(u)\mathop{}\!du\quad\text{and}\quad\sigma_{\alpha}(t)=\int_% {0}^{t}\biggl{(}1-\frac{u}{t}\biggr{)}^{\alpha}f(u)\mathop{}\!dufor {t>0}. The improper integral {\int_{0}^{\infty}f(u)\mathop{}\!du} is said to be {(C,\alpha)} integrable to L for some {\alpha>-1} if the limit {\lim_{x\to\infty}\sigma_{\alpha}(t)=L} exists. It is known that {\lim_{t\to\infty}F(t)=\ell} implies {\lim_{t\to\infty}\sigma_{\alpha}(t)=\ell} for {\alpha>-1}, but the converse of this implication is not true in general. In this paper, we introduce the concept of the general control modulo of non-integer order for functions and obtain some Tauberian conditions in terms of this concept for the {(C,\alpha)} integrability method in order that the converse implication hold true. Our results extend the main theorems in [Ü. Totur and İ. Çanak, Tauberian conditions for the (C,\alpha) integrability of functions, Positivity 21 2017, 1, 73–83].

TAUBERIAN CONDITIONS FOR $q$-CES\`{A}RO INTEGRABILITY

Given a $q$-integrable function $f$ on $[0, \infty)$, we define $s(x)=\int_{0}^{x}f(t)d_qt$ and $\sigma(s(x))=\frac{1}{x}\int _{0}^{x} s(t)d_{q}t$ for $x>0$. It is known that if $\lim _{x \to \infty}s(x)$ exists andis equal to $A$, then $\lim _{x \to \infty}\sigma(s(x))=A$. But the converse of this implication is not true in general. Our goal is to obtain Tauberian conditions imposed on the general control modulo of $s(x)$ under which the converse implication holds. These conditions generalize some previously obtained Tauberian conditions.

On Tauberian theorems for statistical weighted mean method of summability

In this paper we establish some new Tauberian theorems for the statistical weighted mean method of summability via the weighted general control modulo of the oscillatory behavior of nonnegative integer order of a real sequence. The main results improve the well-known classical Tauberian theorems which are given for weighted mean method of summability and statistical convergence.

SOME TAUBERIAN REMAINDER THEOREMS FOR HOLDER SUMMABILITY

In this paper, we prove some Tauberian remainder theorems that generalize the results given by Meronen and Tammeraid [Math. Model. Anal., 18(1):97– 102, 2013] for Holder summability method using the notion of the general control modulo of the oscillatory behaviour of nonnegative integer order.

TAUBERIAN REMAINDER THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY

Using the weighted general control modulo, we prove several Tauberian remainder theorems for the weighted mean method of summability. Our results generalize the results proved by Meronen and Tammeraid [Math. Model. Anal. 18 (1) 2013, 97–102].

GENERAL CONTROL MODULO AND TAUBERIAN REMAINDER THEOREMS FOR (C, 1) SUMMABILITY

We prove for the (C, 1) summability method several Tauberian remainder theorems using the general control modulo of the oscillatory behavior.

Some Tauberian conditions for Cesàro summability method

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: $$\begin{gathered} \left( {\lambda - 1} \right)\mathop {\lim \sup }\limits_n \left( {\frac{1} {{\left[ {\lambda n} \right] - n}}\sum\limits_{k = n + 1}^{\left[ {\lambda n} \right]} {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1} {q}} = o\left( 1 \right), \lambda \to 1^ + , q > 1, \hfill \\ \left( {1 - \lambda } \right)\mathop {\lim \sup }\limits_n \left( {\frac{1} {{n - \left[ {\lambda n} \right]}}\sum\limits_{k = \left[ {\lambda n} \right] + 1}^n {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1} {q}} = o\left( 1 \right), \lambda \to 1^ - , q > 1, \hfill \\ \end{gathered}$$ 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 [DİK, F.: Tauberian theorems for convergence and subsequential convergence with moderately oscillatory behavior, Math. Morav. 5, (2001), 19–56] and [λn] denotes the integer part of λn.

A Tauberian Theorem with a Generalized One-Sided Condition

We prove a Tauberian theorem to recover moderate oscillation of a real sequenceu=(un)out of Abel limitability of the sequence(Vn(1)(Δu))and some additional condition on the general control modulo of oscillatory behavior of integer order ofu=(un).

Tauberian conditions for a general limitable method

Let(un)be a sequence of real numbers,Lan additive limitable method with some property, andanddifferent spaces of sequences related to each other. We prove that if the classical control modulo of the oscillatory behavior of(un)inis a Tauberian condition forL, then the general control modulo of the oscillatory behavior of integer ordermof(un)inoris also a Tauberian condition forL.