scholarly journals Tauberian conditions with controlled oscillatory behavior for statistical convergence

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3853-3865
Author(s):  
Sefa Sezer ◽  
Rahmet Savaş ◽  
İbrahim Çanak
Keyword(s):  
Finite Number ◽  
Statistical Convergence ◽  
Oscillatory Behavior ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Real Sequence ◽  
Tauberian Conditions

We present new Tauberian conditions in terms of the general logarithmic control modulo of the oscillatory behavior of a real sequence (sn) to obtain lim n?? sn = ? from st - lim n?? sn = ?, where ? is a finite number. We also introduce the statistical (l,m) summability method and extend some Tauberian theorems to this method. The main results improve some well-known Tauberian theorems obtained for the 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 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.

New Tauberian Theorems from Old

1994 ◽  
Vol 46 (2) ◽  
pp. 380-394
Author(s):  
Mangalam R. Parameswaran
Keyword(s):  
Absolute Summability ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Tauberian Conditions ◽  
Powerful Approach ◽  
Abstract Spaces

AbstractA new and very general and simple, yet powerful approach is introduced for obtaining new Tauberian theorems for a summability method V from known Tauberian conditions for V, where V is merely assumed to be linear and conservative. The technique yields the known theorems on the weakening of Tauberian conditions due to Meyer-König and Tietz and others and also improves many of them. Several new results are also obtained, even for classical methods of summability, including analogues of Tauber's second theorem for the Borel and logarithmic methods. The approach yields also new Tauberian conditions for the passage from summability by a method V to summability by a method V', as well as to more general methods of summability like absolute summability or summability in abstract spaces; the present paper however confines itself to ordinary summability.

scholarly journals Tauberian conditions for Conull spaces

1985 ◽  
Vol 8 (4) ◽  
pp. 689-692 ◽  
Author(s):  
J. Connor ◽  
A. K. Snyder
Keyword(s):  
Tauberian Theorem ◽  
Convergent Sequence ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Tauberian Conditions

The typical Tauberian theorem asserts that a particular summability method cannot map any divergent member of a given set of sequences into a convergent sequence. These sets of sequences are typically defined by an “order growth” or “gap” condition. We establish that any conull space contains a bounded divergent member of such a set; hence, such sets fail to generate Tauberian theorems for conull spaces.

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

2020 ◽  
pp. 775
Author(s):  
Firat Ozsarac ◽  
Ibrahim Canak
Keyword(s):  
Finite Number ◽  
Weight Function ◽  
Generating Function ◽  
Finite Interval ◽  
Classical Type ◽  
Tauberian Theorems ◽  
Weighted Mean ◽  
Positive Weight ◽  
Tauberian Conditions ◽  
Complex Valued

Let $q$ be a positive weight function on $\mathbf{R}_{+}:=[0, \infty)$ which is integrable in Lebesgue's sense over every finite interval $(0,x)$ for $0<x<\infty$, in symbol: $q \in L^{1}_{loc} (\mathbf{R}_{+})$ such that $Q(x):=\int_{0}^{x} q(t) dt\neq 0$ for each $x>0$, $Q(0)=0$ and $Q(x) \rightarrow \infty $ as $x \to \infty $.Given a real or complex-valued function $f \in L^{1}_{loc} (\mathbf{R}_{+})$, we define $s(x):=\int_{0}^{x}f(t)dt$ and$$\tau^{(0)}_q(x):=s(x), \tau^{(m)}_q(x):=\frac{1}{Q(x)}\int_0^x \tau^{(m-1)}_q(t) q(t)dt\,\,\, (x>0, m=1,2,...),$$provided that $Q(x)>0$. We say that $\int_{0}^{\infty}f(x)dx$ is summable to $L$ by the $m$-th iteration of weighted mean method determined by the function $q(x)$, or for short, $(\overline{N},q,m)$ integrable to a finite number $L$ if$$\lim_{x\to \infty}\tau^{(m)}_q(x)=L.$$In this case, we write $s(x)\rightarrow L(\overline{N},q,m)$. It is known thatif the limit $\lim _{x \to \infty} s(x)=L$ exists, then $\lim _{x \to \infty} \tau^{(m)}_q(x)=L$ also exists. However, the converse of this implicationis not always true. Some suitable conditions together with the existence of the limit $\lim _{x \to \infty} \tau^{(m)}_q(x)$, which is so called Tauberian conditions, may imply convergence of $\lim _{x \to \infty} s(x)$. In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for $(\overline{N},q,m)$ summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Ces\`{a}ro summability $(C,1)$ and weighted mean method of summability $(\overline{N},q)$ have been extended and generalized.  

On Tauberian theorems for (A, k) summability method

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
İbrahim Çanak ◽  
Ümit Totur ◽  
Mehmet Dik
Keyword(s):  
Summability Method ◽  
Tauberian Theorems ◽  
Real Numbers ◽  
Tauberian Conditions

AbstractLet (u n) be a sequence of real numbers. In this paper we introduce some Tauberian conditions in terms of regularly generated sequences for (A, k) summability method.

scholarly journals Some Tauberian conditions on logarithmic density

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Adem Kılıçman ◽  
Stuti Borgohain ◽  
Mehmet Küçükaslan
Keyword(s):  
Statistical Convergence ◽  
Tauberian Theorems ◽  
Logarithmic Density ◽  
Tauberian Conditions ◽  
De La Vallée Poussin

Abstract This article is based on the study on the λ-statistical convergence with respect to the logarithmic density and de la Vallee Poussin mean and generalizes some results of logarithmic λ-statistical convergence and logarithmic $(V,\lambda )$ ( V , λ ) -summability theorems. Hardy’s and Landau’s Tauberian theorems to the statistical convergence, which was introduced by Fast long back in 1951, have been extended by J.A. Fridy and M.K. Khan (Proc. Am. Math. Soc. 128:2347–2355, 2000) in recent years. In this article we try to generalize some Tauberian conditions on logarithmic statistical convergence and logarithmic $(V,\lambda )$ ( V , λ ) -statistical convergence, and we find some new results on it.

Tauberian conditions under which statistical convergence follows from statistical summability by weighted means

2004 ◽  
Vol 41 (4) ◽  
pp. 391-403 ◽  
Author(s):  
Ferenc Móricz ◽  
Cihan Orhan
Keyword(s):  
Finite Number ◽  
Large Class ◽  
Statistical Convergence ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Real Numbers ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Weighted Means ◽  
Necessary And Sufficient

The first named author has recently proved necessary and sufficient Tauberian conditions under which statistical convergence (introduced by H. Fast in 1951) follows from statistical summability (C, 1). The aim of the present paper is to generalize these results to a large class of summability methods (,p) by weighted means. Let p = (pk : k = 0,1, 2,...) be a sequence of nonnegative numbers such that po > 0 and Let (xk) be a sequence of real or complex numbers and set for n = 0,1, 2,.... We present necessary and sufficient conditions under which the existence of the limit st-lim xk = L follows from that of st-lim tn = L, where L is a finite number. If (xk) is a sequence of real numbers, then these are one-sided Tauberian conditions. If (xk) is a sequence of complex numbers, then these are two-sided Tauberian conditions.

scholarly journals Tauberian theorems for the statistical convergence and the statistical (C,1,1) summability

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 101-116 ◽  
Author(s):  
Ümit Totur ◽  
İbrahim Çanak
Keyword(s):  
Statistical Convergence ◽  
Double Sequence ◽  
Convergent Sequence ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Difference Sequence ◽  
Arithmetic Means ◽  
The Difference ◽  
Single Sequence

Every P-convergent double sequence is statistically convergent and every bounded statistically convergent sequence is statistical (C,1,1) summable. The converse of these implications are not always true. Theorems on which conditioned converses are searched are known as Tauberian theorems. Inspired by the convergence to zero of the difference sequence between a sequence and its arithmetic means in the single sequence case, we obtain Tauberian theorems for the statistical convergence and statistical (C,1,1) summability method by imposing some conditions on the difference sequence between a double sequence and its different arithmetic means.

scholarly journals Tauberian theorems for weighted mean statistical summability of double sequences of fuzzy numbers

2021 ◽  
Vol 25 (2) ◽  
pp. 175-187
Author(s):  
Hemen Dutta ◽  
Jyotishmaan Gogoi
Keyword(s):  
Statistical Convergence ◽  
Fuzzy Numbers ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Weighted Mean ◽  
Tauberian Conditions ◽  
Weighted Means ◽  
Sequences Of Fuzzy Numbers

We discuss Tauberian conditions under which the statistical convergence of double sequences of fuzzy numbers follows from the statistical convergence of their weighted means. We also prove some other results which are necessary to establish the main results.

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