TAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS

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 $00$, $Q(0)=0$ 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. 

Tauberian conditions for a general limitable method

Let (un) be a sequence of real numbers, L an additive limitable method with some property, and and different spaces of sequences related to each other. We prove that if the classical control modulo of the oscillatory behavior of (un) in is a Tauberian condition for L, then the general control modulo of the oscillatory behavior of integer order m of (un) in or is also a Tauberian condition for L

A theorem for convergence of generator sequences

WOS: 000287553400026In this paper, we show under which conditions a generator sequence is convergent. We also give a Tauberian theorem for Cesaro summability as a special case. (C) 2010 Elsevier Ltd. All rights reserved

A note on a Tauberian theorem for (A, i) limitable method II

WOS: 000288527200010In this paper which is a sequel to one by Canak and Albayrak [Int. J. Pure Appl. Math. 35 (3) (2007), 421-424], we obtain weaker Tauberian type conditions under which boundedness and subsequential convergence of (u(n)) follows from its (A, i) limitability

Tauberian theorems for Cesaro summability of sequences of fuzzy numbers

WOS: 000341191800034In this paper we use the concept of the Cesaro convergence of a sequence of fuzzy numbers defined by Subrahmanyam [Cesaro summability of fuzzy real numbers, Anal, 7 (1999), 159-168] to prove some Tauberian theorems for sequences of fuzzy numbers and obtain fuzzy analogues of some classical Tauberian theorems for Cesaro summability of sequences of fuzzy numbers as a corollary

On the Riesz mean of sequences of fuzzy real numbers

WOS: 000336491200008We state and prove a fuzzy analogue of a Theorem due to Moricz and Rhoades [Acta Math. Hungar. 102 (4) (2004), 279-285]

On (C, 1) means of sequences

WOS: 000296987000016Let (s(n)) be a sequence of real numbers such that lim sup(n) sigma(n) = beta and lim inf(n) sigma(n) = alpha, where sigma(n) = 1/n Sigma(n)(k=1) s(k) and beta not equal alpha. We prove that lim sup(n) s(n) = beta and lim inf(n) s(n) = alpha if the following conditions hold: lim(n) inf 1/[lambda n] - n Sigma([lambda n])(k=n+1) (s(k) - s(n)) >= (beta - alpha) lambda/lambda - 1 for lambda > 1, lim(n) inf 1/n - [lambda n] Sigma(n)(k=[lambda n]+1) (s(k) - s(k)) >= (beta - alpha) lambda/1 -lambda for 0 < lambda < 1, where [lambda n] denotes the integer part of lambda n. (C) 2011 Elsevier Ltd. All right reserved

On Tauberian theorems for Cesaro summability of sequences of fuzzy numbers

WOS: 000374171500014In this paper, we show under which conditions the limit of the distance between nth term of a sequence of fuzzy numbers and nth term of its Cesaro mean of order one tends to zero. As corollaries we prove several Tauberian theorems for Cesaro summability of sequences of fuzzy numbers

A theorem on the Cesaro summability method

WOS: 000287783400042In this paper we retrieve the backward Cesaro convergence of a real sequence u = (u(n)) from the Cesaro summability of the general control modulo of the oscillatory behavior of integer order m of (u(n)) under certain conditions. (C) 2010 Elsevier Ltd. All rights reserved

A Tauberian Theorem for a Weighted Mean Method of Summability in Ordered Spaces

Canak, Ibrahim/0000-0002-1754-1685WOS: 000516010400005In this paper, we extend a Tauberian theorem for the Cesaro summability method due to Maddox (Analysis 9:297-302, 1989) in ordered spaces to the weighted mean method of summability