Starting from a general sequence of linear and positive operators of discrete type, we associate its r-th order generalization. This construction involves high order derivatives of a signal and it looses the positivity property. Considering that the initial approximation process is A-statistically uniform convergent, we prove that the property is inherited by the new sequence. Also, our result includes information about the uniform convergence. Two applications in q-Calculus are presented. We study q-analogues both of Meyer-König and Zeller operators and Stancu operators.
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
O. Agratini, Statistical convergence of a non-positive approximation process, Chaos, Solitons & Fractals, 44 (2011) no. 11, pp. 977-981, https://doi.org/10.1016/j.chaos.2011.08.003
requires subscription: https://doi.org/10.1016/j.chaos.2011.08.003
About this paper
Chaos, Solitons & Fractals
google scholar link
 Agratini O. On a q-analogue of Stancu operators. Cent Eur J Math 2010;8(1):191–8.
 Altomare F. Korovkin-type theorems and approximation by positive linear operators. Surveys Approx Theory 2010;5:92–164.
 Altomare F, Campiti M. Korovkin-type approximation theory and its applications. de Gruyter studies in mathematics, vol. 17. Berlin: Walter de Gruyter and Co.; 1994.
 Bernstein SN. Démontration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun Soc Math Kharkow 1912–1913:1–2.
 Bohman H. On approximation of continuous and of analytic functions. Ark Mat 1952–1954;2:43–56.
 Connor JS. The statistical and strong p-Cesàro convergence of sequences. Analysis 1988;8:47–63.
 Duman O, Erkus E. Approximation of continuous periodic functions via statistical convergence. Comput Math Appl 2006;52:967–74.
 Duman O, Orhan C. An abstract version of the Korovkin approximation theorem. Publ Math Debrecen 2006;69:33–46. fasc. 1–2.
 Fast H. Sur la convergence statistique. Colloq Math 1951;2:241–4.
 Feller W. An introduction to probability theory and its applications, 2nd ed., vol. I. New York: John Wiley; 1957.
 Fridy JA. On statistical convergence. Analysis 1985;5:301–13.
 Gadjiev AD, Orhan C. Some approximation theorems via statistical convergence. Rocky Mountain J Math 2002;32:129–38.
 Kirov G, Popova L. A generalization of the linear positive operators. Math Balkanica, N.S. 1993;7:149–62. fasc. 2.
 Korovkin PP. On convergence of linear positive operators in the space of continuous functions (Russian). Dokl Akad Nauk SSSR (N.S.) 1953;90:961–4.
 Lorentz GG. Bernstein polynomials. Toronto: Univ. of Toronto Press; 1953.
 Lupas A. A q-analogue of the Bernstein operator, University of ClujNapoca, Seminar on Numerical and Statistical Calculus, vol. 9. Preprint No; 1987. p. 85–92.
 Nowak G. Approximation properties for generalized q-Bernstein polynomials. J Math Anal Appl 2009;350:50–5.
 Ali Özarslan M, Duman O. Approximation theorems by Meyer-König and Zeller type operators. Chaos Solitons Fractals 2009;41:451–6.
 Phillips GM. Bernstein polynomials based on the q-integers. Ann Numer Math 1997;4:511–8.
 Pinkus A. Weierstrass and approximation theory. J Approx Theory 2000;107:1–66.
 Popoviciu T. On the proof of Weierstrass’ theorem using interpolation polynomials (Romanian). Lucrarile Sesiunii Gen. St. Acad. Române, 2-12 iunie 1950, Editura Academiei R.P.R.; 1951. p. 1664–67. [Translated in English by Daniela Kacsó in East Journal on Approximations, 4(1998), fasc. 1, 107–110].
 Stancu DD. Use of probabilistic methods in the theory of uniform approximation of continuous functions. Rev Roum Math Pures Appl 1969;14(5):673–91.
 Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. Colloq Math 1951;2:73–4.
 Zeilberger D. Proof of the refined alternating sign matrix conjecture. New York J Math 1996;2:59–68.