A Korovkin type approximation theorem and its applications. (English) Zbl 1474.41045

Summary: We present a Korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions via \(A\)-statistical approximation, for the rate of the third order Ditzian-Totik modulus of smoothness. Finally, we obtain an interleave between Riesz’s representation theory and Lebesgue-Stieltjes integral-\(i\), for Riesz’s functional supremum formula via statistical limit.


41A36 Approximation by positive operators
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[1] Duman, O., Statistical approximation for periodic functions, Demonstratio Mathematica, 36, 4, 873-878 (2003) · Zbl 1065.41041
[2] Radu, C., \(A\)-summability and approximation of continuous periodic functions, Studia Universitatis, Babeş-Bolyai: Mathematica, 52, 4, 155-161 (2007) · Zbl 1199.41137
[3] Dzyubenko, G. A.; Gilewicz, J., Copositive approximation of periodic functions, Acta Mathematica Hungarica, 120, 4, 301-314 (2008) · Zbl 1164.41003 · doi:10.1007/s10474-008-6204-0
[4] Ditzian, Z.; Totik, V., Moduli of Smoothness (1987), Berlin, Germany: Springer, Berlin, Germany · Zbl 0666.41001
[5] Kopotun, K., On copositive approximation by algebraic polynomials, Analysis Mathematica, 21, 4, 269-283 (1995) · Zbl 0844.41013 · doi:10.1007/BF01909150
[6] Shevchuk, I. A., Approximation by Polynomials and Traces of the Functions Continuous on an Interval (1992 (Russian)), Kiev, Ukraine: Naukova Dumka, Kiev, Ukraine
[7] Korovkin, P. P., Linear Operators and Approximation Theory (1960), Delhi, India: Gordon and Breach, Delhi, India · Zbl 0094.10201
[8] Serfling, R. J., Approximation Theorems of Mathematical Statistics (1980), John Wiley & Sons · Zbl 0538.62002
[9] Sakaoğlu, I.; Ünver, M., Statistical approximation for multivariable integrable functions, Miskolc Mathematical Notes, 13, 2, 485-491 (2012) · Zbl 1274.41051
[10] Rudin, W., Principles of Mathematical Analysis (1976), McGraw-Hill · Zbl 0148.02903
[11] Bartle, R. G., The Elements of Real Analysis (1976), John Wiley & Sons · Zbl 0309.26003
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