scholarly journals A Tauberian Theorem with a Generalized One-Sided Condition

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Ibrahim Çanak ◽  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Additional Condition ◽  
Oscillatory Behavior ◽  
Real Sequence ◽  
General Control ◽  
Integer Order ◽  
General Control Modulo ◽  
Moderate Oscillation

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).

scholarly journals On Tauberian theorems for statistical weighted mean method of summability

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1541-1548 ◽  
Author(s):  
Ümit Totur ◽  
Ibrahim Çanak
Keyword(s):  
Nonnegative Integer ◽  
Statistical Convergence ◽  
Oscillatory Behavior ◽  
Tauberian Theorems ◽  
Real Sequence ◽  
General Control ◽  
Weighted Mean ◽  
Integer Order ◽  
General Control Modulo

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 conditions for Cesàro summability method

2012 ◽  
Vol 62 (2) ◽  
Author(s):  
İbrahi̇m Çanak ◽  
Ümi̇t Totur
Keyword(s):  
Finite Number ◽  
Oscillatory Behavior ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Integer Part ◽  
Real Numbers ◽  
General Control ◽  
Integer Order ◽  
Tauberian Conditions ◽  
General Control Modulo

AbstractLet 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.

scholarly journals Tauberian conditions for a general limitable method

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
İbrahim Canak ◽  
Ümit Totur
Keyword(s):  
Oscillatory Behavior ◽  
Real Numbers ◽  
General Control ◽  
Tauberian Condition ◽  
Integer Order ◽  
Tauberian Conditions ◽  
General Control Modulo ◽  
Classical Control

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.

scholarly journals SOME TAUBERIAN REMAINDER THEOREMS FOR HOLDER SUMMABILITY

2015 ◽  
Vol 20 (2) ◽  
pp. 139-147
Author(s):  
Umit Totur ◽  
Muhammet Ali Okur
Keyword(s):  
Nonnegative Integer ◽  
Summability Method ◽  
Oscillatory Behaviour ◽  
General Control ◽  
Math Model ◽  
Integer Order ◽  
General Control Modulo

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.

scholarly journals On a theorem of W. Meyer-König and H. Tietz

2005 ◽  
Vol 2005 (15) ◽  
pp. 2491-2496 ◽  
Author(s):  
İBrahım Çanak ◽  
Mehmet Dık ◽  
Fılız Dık
Keyword(s):  
Oscillatory Behavior ◽  
Real Numbers ◽  
General Control ◽  
Tauberian Condition ◽  
General Control Modulo ◽  
Classical Control ◽  
Different Order

Let(un)be a sequence of real numbers and letLbe an additive limitable method with some property. We prove that if the classical control modulo of the oscillatory behavior of(un)belonging to some class of sequences is a Tauberian condition forL, then convergence or subsequential convergence of(un)out ofLis recovered depending on the conditions on the general control modulo of the oscillatory behavior of different order.

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

2013 ◽  
Vol 18 (1) ◽  
pp. 97-102 ◽  
Author(s):  
Olga Meronen ◽  
Ivar Tammeraid
Keyword(s):  
Oscillatory Behavior ◽  
Summability Method ◽  
General Control ◽  
General Control Modulo

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

General Tauberian theorems for the Cesàro integrability of functions

2020 ◽  
Vol 27 (4) ◽  
pp. 517-527 ◽  
Author(s):  
Ümit Totur ◽  
İbrahim Çanak
Keyword(s):  
Integrable Function ◽  
Tauberian Theorems ◽  
General Control ◽  
Integer Order ◽  
Improper Integral ◽  
Converse Implication ◽  
Tauberian Conditions ◽  
Locally Integrable Function ◽  
General Control Modulo ◽  
Integrability Of Functions

AbstractFor 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].

A Generalized Tauberian Theorem

1958 ◽  
Vol 10 ◽  
pp. 111-114
Author(s):  
F. R. Keogh ◽  
G. M. Petersen
Keyword(s):  
Tauberian Theorem ◽  
Decimal Place ◽  
Real Sequence ◽  
Decimal Expansion ◽  
First Category ◽  
Dense Set

Let {s(n)} be a real sequence and let x be any number in the interval 0 < x ⩽ 1. Representing x by a non-terminating binary decimal expansion we shall denote by {s(n,x)} the subsequence of {s(n)} obtained by omitting s(k) if and only if there is a 0 in the decimal place in the expansion of x. With this correspondence it is then possible to speak of “a set of subsequences of the first category,” “an everywhere dense set of subsequences,” and so on.

scholarly journals TAUBERIAN REMAINDER THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY

2014 ◽  
Vol 19 (2) ◽  
pp. 275-280 ◽  
Author(s):  
Sefa Anil Sezer ◽  
Ibrahim Canak
Keyword(s):  
General Control ◽  
Weighted Mean ◽  
Math Model ◽  
General Control Modulo

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].

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