$\Lambda^2$-statistical convergence and its application to Korovkin second theorem
DOI:
https://doi.org/10.24193/subbmath.2019.4.08Keywords:
$\Lambda^2-$weighted statistical convergence, Korovkin type theorem, Rate of convergenceAbstract
In this paper, we use the notion of strong $(N, \lambda^2)-$summability to generalize the concept of statistical convergence. We call this new method a $\lambda^2-$statistical convergence and denote by $S_{\lambda^2}$ the set of sequences which are $\lambda^2-$statistically convergent. We find its relation to statistical convergence and strong $(N, \lambda^2)-$summability. We will define a new sequence space and will show that it is Banach space. Also we will prove the second Korovkin type approximation theorem for $\lambda^2$-statistically summability and the rate of $\lambda^2$-statistically summability of a sequence of positive linear operators defined from $C_{2\pi}(\mathbb{R})$ into $C_{2\pi}(\mathbb{R}).$
References
bibitem{A} F. Altomare, Korovkin-type theorems and approximation by positive linear operators,
Survey in Approximation Theory 5 (2010) 92-164.
bibitem{BM} N.L. Braha and T. Mansour, On $Lambda^2$-strong convergence of numerical sequences and Fourier series, Acta Math. Hungar., 141 (1-2) (2013), 113-126.
bibitem{B} N.L.Braha, A new class of sequences related to the $lsb p$ spaces defined by sequences of Orlicz functions. J. Inequal. Appl. 2011, Art. ID 539745, 10 pp.
bibitem{BE} N.L. Braha and Mikail Et, The sequence space $E_{n}^{q}left( M,p,sright) $ and $N_{k}-$ lacunary statistical convergence,
Banach J. Math. Anal. 7 (2013), no. 1, 88-96.
bibitem{BSM} Naim L. Braha, H.M. Srivastava and S.A. Mohiuddine, A Korovkin's type approximation theorem for periodic functions
via the statistical summability of the generalized de la Vallée Poussin mean, Appl. Math. Comput. 228 (2014), 162-169.
bibitem{B1} N.L. Braha, Valdete Loku, H.M. Srivastava, $Lambda^2-$Weighted statistical convergence and Korovkin and Voronovskaya type theorems, Appl. Math. Comput. 266 (2015), 675-686.
bibitem{C} J.S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis 8 (1988), 47-63.
bibitem{EMN} O.H.H. Edely, S.A. Mohiuddine, A.K. Noman, Korovkin type approximation theorems
obtained through generalized statistical convergence, Applied Math. Letters 23 (2010) 1382-1387.
bibitem{F}H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241-244.
bibitem{F1} J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313.
bibitem{FO} J.A. Fridy and C. Orhan, Lacunary statistical convergences, Pacific J. Math. 160 (1993), no. 1, 43-51.
bibitem{LB} Loku, Valdete; Braha, N. L. Some weighted statistical convergence and Korovkin type-theorem. J. Inequal. Spec. Funct. 8 (2017), no. 3, 139-150.
bibitem{K} P.P. Korovkin, Convergence of linear positive operators in the spaces of continuous
functions (Russian). Doklady Akad. Nauk. SSSR (N.S.) 90 (1953) 961-964.
bibitem{K1} P.P. Korovkin, Linear operators and approximation theory, Hindustan Publ. Co.,
Delhi, 1960.
bibitem{MAM1} S.A. Mohiuddine, A. Alotaibi, M. Mursaleen, Statistical summability $(C, 1)$ and a Korovkin type approximation theorem, J. Inequa. Appl. 2012, 2012:172.
bibitem{MA3} M. Mursaleen, A. Alotaibi, Statistical lacunary summability and a Korovkin type approximation theorem, Ann. Univ. Ferrara 57(2) (2011) 373-381.
bibitem{MKEG} M. Mursaleen, V. Karakaya, M. Erturk, F. Gursoy, Weighted statistical convergence
and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218 (2012) 9132-9137.
bibitem{M} M. Mursaleen, $lambda-$ statistical convergences, Math. Slovaca 50 (2000), no. 1, 111-115.
bibitem{SMK} H.M. Srivastava, M. Mursaleen, A. Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Modelling 55 (2012) 2040-2051.
bibitem{S} H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq.
Math. 2 (1951) 73-74.
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