On the core of weighted means of sequences
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On the core of weighted means of sequences Sefa Anıl Sezer1 · ˙Ibrahim Çanak2 Received: 15 April 2019 / Accepted: 1 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract Let ( pn ) be a sequence of nonnegative numbers such that p0 > 0 and Pn := nk=0 pk . The sequence (tn ) of n-th weighted means of a sequence (u n ) is defined by tn :=
n 1 pk u k (n = 0, 1, 2, . . .). Pn k=0
It is well-known from the Knopp’s core theorem that K − cor e(t) ⊆ K − cor e(u) for every real sequence (u n ). But the converse of this inclusion is not true in general. In this paper, we obtain sufficient conditions under which the converse inclusion holds. Keywords Core of a sequence · Weighted mean method of summability · Regularly varying sequence of positive index Mathematics Subject Classification 40A05 · 40E05 · 40C05
1 Introduction Throughout this paper, we denote the spaces of all real sequences and real convergent sequences by w and c, respectively. If A = (ank ) is an infinite matrix with real entries and u = (u n ) is a sequence of real numbers, then Au denotes the transformed sequence whose n-th term is given by (Au)n = ∞ a k=0 nk u k . For two sequence spaces U and V , we say that the matrix A maps U into V if for all u ∈ U , Au exists and belongs to V . By (U , V ) we denote the set of all matrices
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s13370-02000831-z) contains supplementary material, which is available to authorized users.
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˙Ibrahim Çanak [email protected] Sefa Anıl Sezer [email protected]
1
Department of Mathematics, ˙Istanbul Medeniyet University, Istanbul, Turkey
2
Department of Mathematics, Ege University, Izmir, Turkey
123
S. A. Sezer, ˙I. Çanak
which maps U into V . If the limit is preserved then we denote the class of such matrices by (U , V )r eg . A matrix A is said to be regular if A ∈ (c, c)r eg . Let p = ( pn ) be a sequence of nonnegative numbers such that p0 > 0 and Pn :=
n
pk (n = 0, 1, 2, . . .).
k=0
The weighted mean method (N , p) is defined by the infinite matrix (ank ), where (ank ) is given by pk ,0≤k≤n ank = Pn 0, k > n. The sequence t = (tn ) of the n-th weighted mean of a sequence (u n ) is defined by tn := ((N , p)u)n =
n 1 pk u k . Pn k=0
A sequence (u n ) is said to be summable by the weighted mean method determined by the sequence p, in short, (N , p) summable to a finite number L if limn→∞ tn = L. The (N , p) method is regular under the condition Pn :=
n
pk → ∞, n → ∞.
(1)
k=0
(For more details concerning weighted means, we refer the reader to [5, p. 57].) The concept of core (also called Knopp core or briefly K-cor e) of a complex sequence which is inherently related to the set of limit points of the sequence was introduced by Knopp [7]. Let (u n ) be a complex sequence and Rn be the least closed convex region in the complex plane which contains the points u n , u n+1 , u n+2 , . . .. Then, K − cor e(u) =
∞
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