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.

On New Difference Sequence Space and Their Generalised Almost Statistical Convergence

2020 ◽  
pp. 212-219
Author(s):  
Ajaya Kumar Singh
Keyword(s):  
Sequence Space ◽  
Statistical Convergence ◽  
Convergent Sequence ◽  
Difference Sequence ◽  
Almost Convergence ◽  
Topological Properties ◽  
Statistical Limit ◽  
Difference Sequence Space ◽  
Limit Functional ◽  
Convergent Sequences

The object of the present paper is to introduce the notion of generalised almost statistical (GAS) convergence of bounded real sequences, which generalises the notion of almost convergence as well as statistical convergence of bounded real sequences. We also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Also, the existence GAS convergent sequence, which is neither statistical convergent nor almost convergent. Lastly, some topological properties of the space of all GAS convergent sequences are investigated.

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 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.

On regularly generated double sequences

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 809-822 ◽  
Author(s):  
Ümit Totur ◽  
İbrahim Çanak
Keyword(s):  
Double Sequence ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Real Numbers

In this paper, we introduce regularly generated sequences for double sequence of real numbers, and obtain some Tauberian theorems for (C; 1; 1) summability method using the concept of regularly generated sequence.

Almost strongly Orlicz double sequence spaces of regular matrices and their applications to statistical convergence

2018 ◽  
Vol 11 (05) ◽  
pp. 1850073 ◽  
Author(s):  
Kuldip Raj ◽  
Anu Choudhary ◽  
Charu Sharma
Keyword(s):  
Statistical Convergence ◽  
Orlicz Function ◽  
Double Sequence ◽  
Convergent Sequence ◽  
Sequence Spaces ◽  
Regular Matrix ◽  
Normed Spaces ◽  
Double Sequence Spaces ◽  
Inclusion Relations ◽  
Algebraic Properties

In this paper, we introduce and study some strongly almost convergent double sequence spaces by Riesz mean associated with four-dimensional bounded regular matrix and a Musielak–Orlicz function over [Formula: see text]-normed spaces. We make an effort to study some topological and algebraic properties of these sequence spaces. We also study some inclusion relations between the spaces. Finally, we establish some relation between weighted lacunary statistical sequence spaces and Riesz lacunary almost statistical convergent sequence spaces over [Formula: see text]-normed spaces.

Classical Tauberian Theorems for the (C; 1; 1) Summability Method

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Subject Classification ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Littlewood Theorem

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

Confidence intervals for a difference between lognormal means in cluster randomization trials

2014 ◽  
Vol 26 (2) ◽  
pp. 598-614 ◽  
Author(s):  
Julia Poirier ◽  
GY Zou ◽  
John Koval
Keyword(s):  
Confidence Intervals ◽  
Community Acquired Pneumonia ◽  
Small Sample ◽  
Cluster Randomization ◽  
Critical Pathway ◽  
Arithmetic Means ◽  
Multiple Parameters ◽  
The Difference ◽  
Using Data ◽  
Small Sample Sizes

Cluster randomization trials, in which intact social units are randomized to different interventions, have become popular in the last 25 years. Outcomes from these trials in many cases are positively skewed, following approximately lognormal distributions. When inference is focused on the difference between treatment arm arithmetic means, existent confidence interval procedures either make restricting assumptions or are complex to implement. We approach this problem by assuming log-transformed outcomes from each treatment arm follow a one-way random effects model. The treatment arm means are functions of multiple parameters for which separate confidence intervals are readily available, suggesting that the method of variance estimates recovery may be applied to obtain closed-form confidence intervals. A simulation study showed that this simple approach performs well in small sample sizes in terms of empirical coverage, relatively balanced tail errors, and interval widths as compared to existing methods. The methods are illustrated using data arising from a cluster randomization trial investigating a critical pathway for the treatment of community acquired pneumonia.

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.

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.

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