Cent. Eur. J. Math. • 7(3) • 2009 • 558-567 DOI: 10.2478/s11533-009-0009-4 Central European Journal of Mathematics Lacunary equi-statistical convergence of positive linear operators Research Article Hüseyin Aktu ˘ glu 1∗ , Halil Gezer 1† 1 Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Gazima˘ gusa, Turkey Received 26 September 2008; accepted 8 February 2009 Abstract: In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform conver- gence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical con- vergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial ex- tension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied. MSC: 41A36, 40A05,41A25 Keywords: Statistical convergence • Lacunary statistical convergence • A-statistical convergence • Equi-statistical conver- gence • Korovkin type approximation theorem • Order of convergence © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction Let K be a subset of natural numbers and, C 1 be the Cesáro matrix of order one then following Freedman and Sember [8], the asymptotic density of K is given by δ C 1 (K ) = lim n (C 1 χ K (k )) n (1) provided that limit exists. A real sequence x =(x k ) is called statistically convergent to the number L if for each ε> 0, K (ε)= {k ≤ n : |x k − L|≥ ε}) has natural density zero (see [2, 7, 10]). The concept of A−statistical convergence has been initiated by Kolk in [14] by using an arbitrary non-negative regular matrix A instead of C 1 in (1). Recall that a lacunary sequence θ = {k r } , is an increasing integer sequence such that k 0 =0 and h r = k r − k r−1 →∞ as r →∞. Throughout this paper the intervals determined by θ will be denoted by I r =(k r−1 ,k r ] and the ratio kr k r−1 will ∗ E-mail: huseyin.aktuglu@emu.edu.tr † E-mail: halil.gezer@cc.emu.edu.tr 558