Posts by Octavian Agratini


This paper is concerned with a generalization in q-calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.


Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania


q-integers; q-Bernstein polynomials; Uniform convergence; Smoothness; Lipschitz-type maximal function.

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O. Agratini, On a q-analogue of Stancu operators, Cent. Eur. J. Math., 8 (2010) no. 1, pp. 191-198.   DOI: 10.2478/s11533-009-0057-9


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[1] Agratini O., Rus I.A., Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Carolinae, 2003, 44, 555–563

[2] Altomare F., Campiti M., Korovkin-type approximation theory and its applications, de Gruyter Series Studies in Mathematics, vol. 17, Walter de Gruyter & Co., Berlin, New York, 1994

[3] Andrews G.E., q-Series: Their development and application in analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, Conference Board of the Mathematical Sciences, Number 66, American Mathematical Society, 1986

[4] Aral A., A generalization of Szász-Mirakjan operators based on q-integers, Math. Comput. Model., 2008, 47, 1052–1062

[5] Aral A., Dogru O., Bleimann, Butzer and Hahn operators based on the q-integers, J. Ineq. & Appl., 2007, ID 79410

[6] Derriennic M.-M., Modified Bernstein polynomials and Jacobi polynomials in q-calculus, Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl., 2005, 76, 269–290

[7] Dogru O., On statistical approximation properties of Stancu type bivariate generalization of q-Balász-Szabados operators, In: Agratini O., Blaga P. (Eds.), Proc. Int. Conference on Numerical Analysis and Approximation Theory, Cluj-Napoca, Romania, July 5-8, 2006, 179–194, Casa Carţii de Știinţa, Cluj-Napoca, 2006

[8] Il’inskii A., Ostrovska S., Convergence of generalized Bernstein polynomials, J. Approx. Theory, 2002, 116, 100–112

[9] Kac V., Cheung P., Quantum Calculus, Universitext, Springer-Verlag, New York, 2002

[10] Lencze B., On Lipschitz-type maximal functions and their smoothness spaces, Proc. Netherland Acad. Sci. A, 1998,91, 53–63

[11] Lupaș A., A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, Preprint, 1987, 9, 85–92

[12] Lupaș A., q-analogues of Stancu operators, In: Lupaș A., Gonska H., Lupaș L. (Eds.), Mathematical analysis and approximation theory, The 5th Romanian-German Seminar on Approximation Theory and its Applications, RoGer2002, Sibiu, Burg Verlag, 2002, 145–154

[13] Nowak G., Approximation properties for generalized q-Bernstein polynomials, J. Math. Anal. Appl., 2009, 350, 50–55

[14] Ostrovska S., q-Bernstein polynomials and their iterates, J. Approx. Theory, 2003, 123, 232–255

[15] Ostrovska S., The first decade of the q-Bernstein polynomials: Results and perspectives, Journal of Mathematical Analysis and Approximation Theory, 2007, 2, 35–51

[16] Phillips G.M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 1997, 4, 511–518

[17] Phillips G.M., A generalization of the Bernstein polynomials based on the q-integers, Anziam J., 2000, 42, 79–86

[18] Shisha O., Mond B., The degree of convergence of sequences of linear positive operators, Proc. Nat. Acad. Sci. USA,1968, 60, 1196–1200

[19] Stancu D.D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 1968, 8, 1173–1194

[20] Trif T., Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numér. Théor. Approx., 2000, 29,221–229

[21] Videnskii V.S., On some classes of q-parametric positive operators, Operator Theory Adv. Appl., 2005, 158, 213–222

[22] Wang H., Voronovskaja type formulas and saturation of convergence for q-Bernstein polynomials for 0< q <1, J. Approx. Theory, 2007, 145, 182–195

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Abstract This paper is concerned with a generalization in q-calculus of Stancu operators. Involving modulus of continuity and Lipschitz type…