A Tauberian Theorem for a Weighted Mean Method of Summability in Ordered Spaces
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A Tauberian Theorem for a Weighted Mean Method of Summability in Ordered Spaces I˙brahim C ¸ anak1
Received: 8 August 2018 / Revised: 12 January 2020 / Accepted: 18 January 2020 Ó The National Academy of Sciences, India 2020
Abstract In this paper, we extend a Tauberian theorem for the Cesa`ro summability method due to Maddox (Analysis 9:297–302, 1989) in ordered spaces to the weighted mean method of summability. Keywords Tauberian theorem Weighted mean method of summability Uniform weighted mean method of summability
Let p ¼ ðpn Þ be a sequence of nonnegative numbers such P that p0 [ 0, Pn :¼ nk¼0 pk and k :¼ lim
n!1
P½an [1 Pn
ð1Þ
for every a [ 1. Here, ½an denotes the integer part of the product an. In particular, it follows that Pn ! 1; n ! 1: The nth weighted means of ðsn Þ are defined by n 1 X pk s k : rn ¼ ð2Þ Pn k¼0 If the limit lim sn ¼ L
n!1
ð3Þ
exists, then the limit lim rn ¼ L
n!1
ð4Þ
also exists. In general, the converse implication is not true in general.
& I˙brahim C¸anak [email protected] 1
Department of Mathematics, Ege University, ˙Izmir, Turkey
Note that (4) implies (3) under certain conditions imposed on the sequence ðsn Þ and the weight sequence p, which is called a Tauberian condition. Any theorem which states that convergence of sequences follows from its summability by the weighted mean method summability determined by the sequence p and some Tauberian condition(s) is said to be a Tauberian theorem for the weighted mean method of summability. In the real case, one-sided Tauberian theorem for weighted P mean method of summability states that if a series ak ¼ a0 þ a1 þ of real numbers is summable by weighted mean method to some number s, condition (1) holds and if there exists N and a positive P constant H such Pk1 that pk ak H for all k [ N, then ak converges to s. P More generally, it is shown in [1] that if ak is summable by weighted mean method to some number s, condition (1) holds and ðsk Þ ¼ ða0 þ a1 þ þ ak Þ is slowly P decreasing in the sense of Schmidt [2], then ak converges to s. In this paper, we extend a Tauberian theorem for the Cesa`ro summability method in [3] to the weighted mean method of summability in an abstract ordered linear space. Throughout, we consider an ordered linear space ðX; Þ over the real numbers, in which by 0 we denote the zero element. The following two definitions are cited in [3]. A sequence ðxn Þ in X is said to converge to x 2 X, relative to a given nonnegative q 2 X, if and only if: for all [ 0; there exists n0 [ 0 such that q xn x q whenever n [ n0 . A sequence ðsn Þ in X is said to be slowly decreasing, relative to a given nonnegative q 2 X, if and only if: for all [ 0; there exists n1 [ 0 and there exists d [ 0 such that
123
I˙. C¸anak
sm sn q;
cn ¼
whenever n1 \n\m nð1 þ dÞ. In a general ordered linear space ðX; Þ, we consider a P given series 1 n¼0 an with its sequence of partial sums ðsn Þ. The nth weighted means of ðsn Þ are defined by n 1 X rn ¼ pk
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