On integral type generalizations of positive linear operators

O. Duman; M. A. Özarslan; O. Doğru

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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 convergence. 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 convergence are equivalent to each other. A Korovkin...

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The pointwise approximation properties of the Bézier variant of the MKZ-Kantorovich operators ${\hat{M}}_{n, α} (f, x)$ for α ≥ 1 have been studied in [Comput. Math. Appl., 39 (2000), 1-13]. The aim of this paper is to deal with the pointwise approximation of the operators ${\hat{M}}_{n, α} (f, x)$ for the other case 0 < α < 1. By means of some new techniques and new inequalities we establish an estimate formula on the rate of convergence of the operators ${\hat{M}}_{n, α} (f, x)$ for the case 0 < α < 1. In the end we propose the q-analogue of...