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 |
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.