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2010 | 8 | 1 | 191-198
Tytuł artykułu

On a q-analogue of Stancu operators

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
8
Numer
1
Strony
191-198
Opis fizyczny
Daty
wydano
2010-02-01
online
2010-02-02
Twórcy
Bibliografia
  • [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 http://dx.doi.org/10.1016/j.mcm.2007.06.018
  • [5] Aral A., Doğru 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] Doğru 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 CăŢii de ştiinŢă, Cluj-Napoca, 2006
  • [8] Il’inskii A., Ostrovska S., Convergence of generalized Bernstein polynomials, J. Approx. Theory, 2002, 116, 100–112 http://dx.doi.org/10.1006/jath.2001.3657
  • [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] Lupaş A., A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, Preprint, 1987, 9, 85–92
  • [12] Lupaş A., q-analogues of Stancu operators, In: Lupaş A., Gonska H., Lupaş L. (Eds.), Mathematical analysis and approximation theory, The 5th Romanian-German Seminar on Approximation Theory and its Applications, RoGer 2002, Sibiu, Burg Verlag, 2002, 145–154
  • [13] Nowak G., Approximation properties for generalized q-Bernstein polynomials, J. Math. Anal. Appl., 2009, 350, 50–55 http://dx.doi.org/10.1016/j.jmaa.2008.09.003
  • [14] Ostrovska S., q-Bernstein polynomials and their iterates, J. Approx. Theory, 2003, 123, 232–255 http://dx.doi.org/10.1016/S0021-9045(03)00104-7
  • [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 http://dx.doi.org/10.1017/S1446181100011615
  • [18] Shisha O., Mond B., The degree of convergence of sequences of linear positive operators, Proc. Nat. Acad. Sci. USA, 1968, 60, 1196–1200 http://dx.doi.org/10.1073/pnas.60.4.1196
  • [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 http://dx.doi.org/10.1007/3-7643-7340-7_15
  • [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 http://dx.doi.org/10.1016/j.jat.2006.08.005
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-009-0057-9
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