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    Delta-Quasi-slowly oscillating continuity
    (ScienceDirect, 2016) Çakallı, Hüseyin; Çanak, İbrahim; Dik, Mehmet
    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!1? limn max n?16k6½kn 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 G-convergent [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 G-convergent with G(x) = lim x. A method G is called subsequential if whenever x is G-convergent with G(x) = ‘, then there is a subsequence ?xnk ? of x with limkxnk ¼ ‘. A function f is called G-continuous [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].
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    On tauberian theorems for (A, k) summability method
    (Maltepe Üniversitesi, 2009) Çanak, İbrahim; Totur, Ümit; Dik, Mehmet
    Let (un) be a sequence of real numbers which is (A, k) summable. In this work, several new Tauberian theorems for (A, k) summability methods will be given in terms of generating sequences of (un).