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Lacunary equi-statistical convergence of positive linear operators. (English) Zbl 1181.41039
Summary: 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 type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence are studied by the help of modulus of continuity of positive linear operators.

41A36Approximation by positive operators
40A05Convergence and divergence of series and sequences
41A25Rate of convergence, degree of approximation
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