Gupta, Vijay Some approximation properties of \(q\)-Durrmeyer operators. (English) Zbl 1142.41008 Appl. Math. Comput. 197, No. 1, 172-178 (2008). Summary: We introduce a simple \(q\) analogue of well known Durrmeyer operators. We first estimate moments of \(q\)-Durrmeyer operators. We also establish the rate of convergence for \(q\)-Durrmeyer operators. Cited in 1 ReviewCited in 97 Documents MSC: 41A36 Approximation by positive operators 33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\) Keywords:\(q\)-factorial; \(q\)-integers; \(q\)-Durrmeyer operators; \(q\)-beta function; modulus of continuity PDFBibTeX XMLCite \textit{V. Gupta}, Appl. Math. Comput. 197, No. 1, 172--178 (2008; Zbl 1142.41008) Full Text: DOI References: [1] Derriennic, M. M., Sur ℓ approximation de fonctions integrables sur [0,1] par des polynomes de Bernstein modifies, J. Approx. Theory, 31, 325-343 (1981) · Zbl 0475.41025 [2] Derriennic, M. M., Modified Bernstein polynomials and Jacobi polynomials in \(q\)-calculus, Rend. Circ. Mat. Palermo, Serie II, Suppl. 76, 269-290 (2005) · Zbl 1142.41002 [3] II’inskii, A.; Ostrovska, S., Convergence of generalized Bernstein polynomials, J. Approx. Theory, 116, 1, 100-112 (2002) · Zbl 0999.41007 [4] Kac, V.; Cheung, P., Quantum Calculus (2002), Springer: Springer New York · Zbl 0986.05001 [5] Lorentz, G. G., Bernstein Polynomials, Math. Expo., vol. 8 (1953), Univ. of Toronto Press: Univ. of Toronto Press Toronto · Zbl 0051.05001 [6] Ostrovska, S., \(q\)-Bernstein polynomials and their iterates, J. Approx. Theory, 123, 2, 232-255 (2003) · Zbl 1093.41013 [7] Ostrovska, S., The first decade of the \(q\)-Bernstein polynomials: results and perspectives, J. Math. Anal. Approx. Theory, 2, 1, 35-51 (2007) · Zbl 1159.41301 [8] Phillips, G. M., Bernstein polynomials based on the \(q\)-integers, Ann. Numer. Math., 4, 511-518 (1997) · Zbl 0881.41008 [9] Phillips, G. M., Interpolation and Approximation by Polynomials. Interpolation and Approximation by Polynomials, CMS Books in Math, vol. 14 (2003), Springer · Zbl 1023.41002 [10] Thomae, J., Beitrage zur Theorie der durch die Heinsche Reihe, J. Reine. Angew. Math., 70, 258-281 (1869) · JFM 02.0122.04 [11] Videnskii, V. S., On some classes of \(q\)-parametric positive operators, Operator Theory: Adv. Appl., 158, 213-222 (2005) · Zbl 1088.41008 [12] Wang, H., Korovkin-type theorem and application, J. Approx. Theory, 132, 2, 258-264 (2005), 213-222 · Zbl 1118.41015 [13] Wang, H., Voronovskaya-type formulas and saturation of convergence for \(q\)-Bernstein polynomials for \(0<q<1\), J. Approx. Theory, 145, 182-195 (2007) · Zbl 1112.41016 [14] H. Wang, Properties of convergence for \(Ω; q\); H. Wang, Properties of convergence for \(Ω; q\) [15] Wang, H.; Meng, F., The rate of convergence of \(q\)-Bernstein polynomials for \(0<q<1\), J. Approx. Theory, 136, 2, 151-158 (2005) · Zbl 1082.41007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.