On certain Durrmeyer type \(q\) Baskakov operators. (English) Zbl 1205.41014
Summary: We introduce the \(q\) analogue of certain Durrmeyer type Baskakov operators. These operators were first considered by Z. Finta [J. Math. Anal. Appl. 312, No. 1, 159–180 (2005; Zbl 1079.41019)]. We establish direct results in terms of modulus of continuity.
MSC:
| 41A25 | Rate of convergence, degree of approximation |
| 41A35 | Approximation by operators (in particular, by integral operators) |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
Keywords:
\(q\)-Baskakov operators; Durrmeyer type \(q\) Baskakov operators; modulus of continuity; \(K\)-functionalCitations:
Zbl 1079.41019References:
| [1] | Aral A., Gupta V.: On -Baskakov type operators. Demonstr. Math. 42(1), 109-122 (2009) · Zbl 1176.41028 |
| [2] | Aral A., Gupta V.: On the Durrmeyer type modification of the q Baskakov type operators. Nonlinear Anal. Theory Methods Appl. 72(3-4), 1171-1180 (2010) · Zbl 1180.41012 |
| [3] | Radu C.: On statistical approximation of a general class of positive linear operators extended in q-calculus. Appl. Math. Comput. 215, 2317-2325 (2009) · Zbl 1179.41025 · doi:10.1016/j.amc.2009.08.023 |
| [4] | De Vore R.A., Lorentz G.G.: Constructive Approximation. Springer, Berlin (1993) · Zbl 0797.41016 |
| [5] | Finta Z.: On converse approximation theorems. J. Math. Anal. Appl. 312, 159-180 (2005) · Zbl 1079.41019 · doi:10.1016/j.jmaa.2005.03.044 |
| [6] | Gasper G., Rahman M.: Basic hypergeometrik Series, Encyclopedia of Mathematics and its Applications, vol. 35. Cambridge University Press, Cambridge (1990) · Zbl 0695.33001 |
| [7] | Gupta V.: Approximation for modified Baskakov Durrmeyer type operators. Rocky Mt. J. Math. 39(3), 825-841 (2009) · Zbl 1172.41005 · doi:10.1216/RMJ-2009-39-3-825 |
| [8] | Jackson F.H.: On a q-definite integrals. Q. J. Pure Appl. Math. 41, 193-203 (1910) · JFM 41.0317.04 |
| [9] | Kac V.G., Cheung P.: Quantum Calculus, Universitext. Springer, New York (2002) · Zbl 0986.05001 |
| [10] | Koornwinder, T.H.: q-Special functions, a tutorial. In: Gerstenhaber, M., Stasheff, J. (eds.) Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Contemp. Math. 134, Am. Math. Soc. (1992) · Zbl 0768.33018 |
| [11] | De Sole A., Kac V.G.: On integral representations of q-gamma and q-betta functions. Rend. Mat. Acc. Lincei, s. 9, USA 16(1), 11-29 (2005) · Zbl 1225.33017 |
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