Strongly Statistical Convergence
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STRONGLY STATISTICAL CONVERGENCE U. Kaya1,2 and N. D. Aral3
UDC 519.21
We introduce a concept of A-strongly statistical convergence for sequences of complex numbers, where infinite matrix with nonnegative entries. A sequence (xn ) is called strongly conA = (ank )n,k2N is anX 1 ank |xk − L| = 0 in the ordinary sense. In the definition of A-strongly stavergent to L if lim n!1
k=1
tistical limit, we use the statistical limit instead of the ordinary limit with a convenient density. We study some densities and show that the (ank )-strongly statistical limit is an (amn k )-strong limit, where the density of the set {mn 2 N : n 2 N} is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence (rn ) is dense positive provided the limit superior of a subsequence (rmn ) is positive for all (mn ) with density equal to 1. We show that the dense positivity sufficient condition for the uniqueness of A-strongly statistical limit, where of (rn ) is a necessary and X1 A = (ank ) and rn = ank . Furthermore, necessary conditions for the regularity, linearity and k=1 multiplicativity of the A-strongly statistical limit are established.
1. Introduction The ordinary concept of limit has numerous useful applications in several fields of mathematics, statistics, physics, engineering, and so on. It is well known that a complex sequence is convergent to a point if and only if every neighborhood of this point includes all elements of the sequence except finitely many. If a sequence (xn ) converges to L, then we write lim |xn − L| = 0.
n!1
(1.1)
Hamilton and Hill [8] developed this concept by introducing strong summability in 1938. They generalized equality (1.1) as follows: lim
n!1
1 X k=1
ank |xk − L|p = 0,
(1.2)
where A = (ank ) is an infinite matrix and p > 0. If equality (1.2) holds, then (xn ) is said to be strongly summable to L. In the case where A is the identity matrix and p = 1 in (1.2), we get the ordinary convergence in (1.1). Whenever a new convergence method is introduced, mathematicians investigate its typical properties, such as the uniqueness of limit point, regularity, linearity, etc. Under certain conditions, Hamilton and Hill studied these typical properties of strong convergence. 1
Bitlis Eren University, Bitlis, Turkey; e-mail: [email protected]. Corresponding author. 3 Bitlis Eren University, Bitlis, Turkey; e-mail: [email protected]. 2
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 221–231, February, 2020. Original article submitted December 25, 2016; revision submitted August 3, 2017. 246
0041-5995/20/7202–0246
© 2020
Springer Science+Business Media, LLC
S TRONGLY S TATISTICAL C ONVERGENCE
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In 1963, Wlodarski [15] generalized strong summability to the strong continuous summability method. He studied a sequence of continuous functions (ak (t)) instead of an infinite matrix (ank ) and gave his definition as follows: 1 X lim ak (t)|xk − L|p = 0. t!T
k=1
In addition, he defined some pseudonormed, normed, and Banach spa
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