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