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Volume 27, Issue 1
Approximation Properties of rth Order Generalized Bernstein Polynomials Based on $q$-Calculus

H. Sharma

Anal. Theory Appl., 27 (2011), pp. 40-50.

Published online: 2011-01

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  • Abstract

In this paper we introduce a generalization of Bernstein polynomials based on $q$ calculus. With the help of Bohman-Korovkin type theorem, we obtain $A$−statistical approximation properties of these operators. Also, by using the Modulus of continuity and Lipschitz class, the statistical rate of convergence is established. We also gives the rate of $A$−statistical convergence by means of Peetre’s type $K$−functional. At last, approximation properties of a rth order generalization of these operators is discussed.

  • AMS Subject Headings

41A25, 41A35

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COPYRIGHT: © Global Science Press

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@Article{ATA-27-40, author = {H. Sharma}, title = {Approximation Properties of rth Order Generalized Bernstein Polynomials Based on $q$-Calculus}, journal = {Analysis in Theory and Applications}, year = {2011}, volume = {27}, number = {1}, pages = {40--50}, abstract = {

In this paper we introduce a generalization of Bernstein polynomials based on $q$ calculus. With the help of Bohman-Korovkin type theorem, we obtain $A$−statistical approximation properties of these operators. Also, by using the Modulus of continuity and Lipschitz class, the statistical rate of convergence is established. We also gives the rate of $A$−statistical convergence by means of Peetre’s type $K$−functional. At last, approximation properties of a rth order generalization of these operators is discussed.

}, issn = {1573-8175}, doi = {https://doi.org/10.1007/s10496-011-0040-8}, url = {http://global-sci.org/intro/article_detail/ata/4578.html} }
TY - JOUR T1 - Approximation Properties of rth Order Generalized Bernstein Polynomials Based on $q$-Calculus AU - H. Sharma JO - Analysis in Theory and Applications VL - 1 SP - 40 EP - 50 PY - 2011 DA - 2011/01 SN - 27 DO - http://doi.org/10.1007/s10496-011-0040-8 UR - https://global-sci.org/intro/article_detail/ata/4578.html KW - $q$−integers, $q$−Bernstein polynomials, $A$−statistical convergence, modulus of continuity, Lipschitz class, Peetre’s type $K$−functional. AB -

In this paper we introduce a generalization of Bernstein polynomials based on $q$ calculus. With the help of Bohman-Korovkin type theorem, we obtain $A$−statistical approximation properties of these operators. Also, by using the Modulus of continuity and Lipschitz class, the statistical rate of convergence is established. We also gives the rate of $A$−statistical convergence by means of Peetre’s type $K$−functional. At last, approximation properties of a rth order generalization of these operators is discussed.

H. Sharma. (2011). Approximation Properties of rth Order Generalized Bernstein Polynomials Based on $q$-Calculus. Analysis in Theory and Applications. 27 (1). 40-50. doi:10.1007/s10496-011-0040-8
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