DeltaQuasislowly oscillating continuity
Citation
Çakallı, H., Çanak, İ. ve Dik, M. (2010). Deltaquasislowly oscillating continuity. Applied Mathematics and Computation. s. 28652868.Abstract
Firstly, some definitions and notations will be given in the following. Throughout this paper, N will denote the set of all
positive integers. We will use boldface letters x,y,z,. . . for sequences x = (xn),y = (yn),z= (zn), . . . of terms in R, the set of all
real numbers. Also, s and c will denote the set of all sequences of points in R and the set of all convergent sequences of points
in R, respectively.
A sequence x = (xn) of points in R is called statistically convergent [1] to an element ‘ of R if
lim
n!1
1
n jfk 6 n : jxk ‘j P egj ¼ 0;
for every e > 0, and this is denoted by st limn?1xn = ‘.
A sequence x = (xn) of points in R is slowly oscillating [2], denoted by x 2 SO, if
lim
k!1þ
limn max
nþ16k6½kn
jxk xnj ¼ 0;
where [kn] denotes the integer part of kn. This is equivalent to the following: xm xn?0 whenever 1 6 mn
! 1 as m,n?1.
In terms of e and d, this is also equivalent to the case when for any given e > 0, there exist d = d (e) > 0 and a positive integer
N = N(e) such that jxm xnj < e if nPN(e) and n 6 m 6 (1 + d)n.
By a method of sequential convergence, or briefly a method, we mean a linear function G defined on a sublinear space of s,
denoted by cG(R), into R. A sequence x = (xn) is said to be Gconvergent [3] to ‘ if x 2 cG(R) and G(x) = ‘. In particular, lim
denotes the limit function lim x = limnxn on the linear space c. A method G is called regular if every convergent sequence
x = (xn) is Gconvergent with G(x) = lim x. A method G is called subsequential if whenever x is Gconvergent with G(x) = ‘,
then there is a subsequence ðxnk Þ of x with limkxnk ¼ ‘. A function f is called Gcontinuous [3] if G(f(x)) = f (G(x)) for any Gconvergent
sequence x. Here we note that for special G = st lim, f is called statistically continuous [3]. For real and complex
number sequences, we note that the most important transformation class is the class of matrix methods. For more information
for classical and modern summability methods see [4].
Source
Applied Mathematics and ComputationVolume
216URI
https://www.sciencedirect.com/search?docId=00963003&title=Deltaquasislowly%20oscillating%20continuityhttps://hdl.handle.net/20.500.12415/6165
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 Makale Koleksiyonu [586]
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