A summability process on Baskakov-type approximation
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A summability process on Baskakov-type approximation ˙ Ismail Aslan1 · Oktay Duman1
© Akadémiai Kiadó, Budapest, Hungary 2016
Abstract The summability process introduced by Bell (Proc Am Math Soc 38: 548–552, 1973) is a more general and also weaker method than ordinary convergence. Recent studies have demonstrated that using this convergence in classical approximation theory provides many advantages. In this paper, we study the summability process to approximate a function and its derivatives by means of a wider class of linear operators than a family of positive linear operators. Our results improve not only Baskakov’s idea in (Mat Zametki 13: 785–794, 1973) but also the Korovkin theory based on positive linear operators. In order to verify this we display a specific sequence of approximating operators by plotting their graphs. Keywords
Summability process · Baskakov-type approximation · The Korovkin theory
Mathematics Subject Classification
40C05 · 40D05 · 40G05 · 41A36
1 Introduction By Baskakov’s results in 1973 [5], it is possible to find a wider class of linear approximating operators than a family of positive linear operators of the classical Korovkin theory [1,14]. In recent years some convergence methods, such as, almost convergence, statistical convergence, arithmetic mean convergence, summability processes have been used in Korovkin theory in order to overcome, in many cases, the lack of the usual convergence (see, for instance, [3,4, 11,17,18,20]). Also, statistical analogs of Baskakov’s results have recently been examined by Anastassiou and Duman [2]. The main purpose of the present work, as in the paper by Swetits on Korovkin theory (see [20]), is to study Baskakov-type approximation via the concept of
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Oktay Duman [email protected]; [email protected] ˙Ismail Aslan [email protected]
1
Department of Mathematics, TOBB Economics and Technology University, 06530 Sö˘gütözü, Ankara, Turkey
123
˙I. Aslan, O. Duman
the summability process which was first introduced by Bell [6]. In the next section we provide applications and remarks to demonstrate why such a summability process is needed in approximation theory. We begin with recalling some notations and definitions. Bell [6] defined a summability υ process υ (method) A the following way. Let a sequence of infinite matrices A = {A } = ank (k, n, υ ∈ N) and a sequence of real numbers x = {x k } be given. Then, we say that the double sequence Ax = {(Ax)υn } given by ∞ υ υ ank xk (n, υ ∈ N) (Ax)n := k=1
is a A-transform of x whenever the series converges for every n, υ ∈ N. In this case, a sequence x is called A-summable to some number L provided that lim (Ax)υn = L ,
n→∞
uniformly in υ ∈ N.
(1.1)
We also say that A is regular whenever limk→∞ xk = L implies (1.1). The following characυ terization for regularity was proved by Bell [6]: “A = ank is regular ∞ ifυand only if (a) for υ = 0, uniformly in υ ∈ N; (b) lim each k ∈ N, limn→∞ ank n→∞ k=1 ank = 1, uniformly ∞ υ
a < ∞, and there exist integers N and M such in υ ∈ N;
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