Korovkin type approximation theorem for functions of two variables through statistical A -summability
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RESEARCH
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Korovkin type approximation theorem for functions of two variables through statistical A-summability Mohammad Mursaleen1* and Abdullah Alotaibi2 * Correspondence: [email protected] 1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Full list of author information is available at the end of the article
Abstract In this article, we prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical A-summability. We also study the rate of statistical A-summability of positive linear operators. Finally we construct an example by Bleimann et al. operators to show that our result is stronger than those of previously proved by other authors. AMS Subject Classification 2000: 41A10; 41A25; 41A36; 40A30; 40G15. Keywords: density, statistical convergence, A-statistical convergence, statistical Asummability, positive linear operator, Korovkin type approximation theorem
1 Introduction and preliminaries The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and further studied many others. Let K ⊆ N and Kn = {k ≤ n : k Î K}. Then the natural density of K is defined by δ(K) = limn n-1 |Kn| if the limit exists, where |Kn| denotes the cardinality of Kn. A sequence x = (xk) of real numbers is said to be statistically convergent to L provided that for every ε >0 the set Kε:={kÎN:|xk-L|≥ε} has natural density zero, i.e. for each ε > 0, 1 lim |{k ≤ n : |xk − L| ≥ ε}| = 0. n n
In this case we write st- lim x = L. Note that if x = (xk) is convergent then it is statistically convergent but not conversely. The idea of statistical convergence of double sequences has been intruduced and studied in [2,3]. Let A = (ank), n, kÎN, be an infinite matrix and x = (xk) be a sequence. Then the (transformed) sequence, Ax := (yn), is denoted by yn :=
∞
ank xk ,
k=1
where it is assumed that the series on the right converges for each nÎN. We say that a sequence x is A-summable to the limit ℓ if yn ® ℓ as n ® ∞. A matrix transformation is said to be regular if it maps every convergent sequence into a convergent sequence with the same limit. The well-known conditions for two dimensional matrix to be regular are known as Silverman-Toeplitz conditions.
© 2012 Mursaleen and Alotaibi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Mursaleen and Alotaibi Advances in Difference Equations 2012, 2012:65 http://www.advancesindifferenceequations.com/content/2012/1/65
In [4], Edely and Mursaleen have given the notion of statistical A-summability for single sequences and statistical A-summability for double sequences has recently been studied in [5]. Let A = (a nk ) be a nonnegative regular summability matrix and x = (x k ) be a sequence of real or complex sequences. We say that
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