APPROXIMATION THEOREMS FOR LIMIT $(p,q)-$BERNSTEIN-DURRMEYER OPERATOR

Zoltan Finta, Vijay Gupta

Abstract


In the present paper, using the method developed in \cite{Finta1}, we prove the existence of the limit operator of the slight modification of the sequence of $(p,q)$-Bernstein-Durrmeyer operators introduced recently in \cite{Gupta1}. We also establish the rate of convergence of this limit operator.

Keywords


$(p,q)$-integers, $(p,q)$-Bernstein-Durrmeyer operators, limit $(p,q)$-Bernstein-Durrmeyer operator, rate of convergence, modulus of continuity.

Full Text:

PDF

References


bibitem{Aral}

Aral, A., Gupta, V., Agarwal, R. P.:

{em Applications of $q$-Calculus in Operator Theory,}

Springer, New York, 2013.

bibitem{Derriennic}

Derriennic, M. M.:

{em Sur l'approximation de fonctions int'egrables sur $[0,1]$ par des polyn^omes de Bernstein modifies,}

J. Approx. Theory, 31 (1981), 325-343.

bibitem{Finta1}

Finta, Z,:

{em Korovkin type theorem for sequences of operators depending on a parameter,}

Demonstratio Math., 48(3)(2015), 391-403.

bibitem{Finta2}

Finta, Z.:

{em Approximation by limit $q-$Bernstein operator,}

Acta Univ. Sapientiae, Mathematica, 5(1)(2013), 39-46.

bibitem{Gupta0}

Gupta, V.:

{em (p,q)-Sz'asz-Mirakyan-Baskakov operators,}

Complex Anal. Operator Theory, (in press), DOI

1007/s11785-015-0521-4.

bibitem{Gupta1}

Gupta, V., Aral, A.:

{em Bernstein Durrmeyer operators based on two parameters,}

Facta Universitatis (Niu s), Ser. Math. Inform31(1) (2016), 79-95.

bibitem{Gupta2}

Gupta, V., Agarwal, R. P.:

{em Convergence Estimates in Approximation Theory,}

Springer, New York, 2014.

bibitem{Ilinskii}

Il'inskii, A., Ostrovska, S.:

{em Convergence of generalized Bernstein polynomials,}

J. Approx. Theory, 116 (2002), 100-112.

bibitem{Lupas}

Lupac s, A.,

{em A $q$-analogue of the Bernstein operator,}

Babec s-Bolyai University, Seminar on Numerical and Statistical Calculus, 9 (1987), 85-92.

bibitem{gvm} Milovanovic, G. V., Gupta V., Malik, N., $(p,q)$-Beta functions and applications in approximation, Boletín de la Sociedad Matemática Mexicana, in press DOI

1007/s40590-016-0139-1

bibitem{Mursaleen}

Mursaleen, M., Ansari, K. J., Khan, A.:

{em On (p,q)-analogue of Bernstein operators,}

Appl. Math. Comput., 266 (2015), 874-882.

bibitem{Oruc}

Oruc c, H., Phillips, G. M.:

{em A generalization of the Bernstein polynomials,}

Proc. Edinb. Math. Soc., 42 (1999), 403-413.

bibitem{Phillips1}

Phillips, G. M.,

{em Bernstein polynomials based on the $q$-integers,}

Ann. Numer. Math., 4 (1997), 511-518.

bibitem{Sadjang1}

Sadjang, P. N.:

{em On the $(p,q)$-Gamma and the $(p,q)$-Beta functions,}

arXiv:1506.0739v1.22Jun2015.

bibitem{Sadjang2}

Sadjang, P. N.:

{em On the fundamental theorem of $(p,q)$-calculus and some $(p,q)$-Taylor formulas,}

arXiv:1309.3934[math.QA].

bibitem{Sahai}

Sahai, V., Yadov, S.:

{em Representations of two parameter quantum algebras and $p,q$-special functions,}

J. Math. Anal. Appl., 335 (2007), 268-279.

bibitem{Wang}

Wang, H., Meng, F.,

{em The rate of convergence of $q-$Bernstein polynomials for $0

J. Approx. Theory, 136 (2005), 151-158.


Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)