Approximation properties of bivariate complex q -Bernstein polynomials in the case q > 1

Nazim I. Mahmudov

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 2, page 557-566
  • ISSN: 0011-4642

Abstract

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In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate q -Bernstein polynomials for a function analytic in the polydisc D R 1 × D R 2 = { z ∈ C : | z | < R 1 } × { z ∈ C : | z | < R 1 } for arbitrary fixed q > 1 . We give quantitative Voronovskaja type estimates for the bivariate q -Bernstein polynomials for q > 1 . In the univariate case the similar results were obtained by S. Ostrovska: q -Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.

How to cite

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Mahmudov, Nazim I.. "Approximation properties of bivariate complex $q$-Bernstein polynomials in the case $q>1$." Czechoslovak Mathematical Journal 62.2 (2012): 557-566. <http://eudml.org/doc/246993>.

@article{Mahmudov2012,
abstract = {In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate $q$-Bernstein polynomials for a function analytic in the polydisc $D_\{R_\{1\}\}\times D_\{R_\{2\}\}=\lbrace z\in C\colon \vert z\vert <R_\{1\}\rbrace \times \lbrace z\in C\colon \vert z\vert <R_\{1\}\rbrace $ for arbitrary fixed $q>1$. We give quantitative Voronovskaja type estimates for the bivariate $q$-Bernstein polynomials for $q>1$. In the univariate case the similar results were obtained by S. Ostrovska: $q$-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.},
author = {Mahmudov, Nazim I.},
journal = {Czechoslovak Mathematical Journal},
keywords = {$q$-Bernstein polynomials; modulus of continuity; Voronovskaja type theorem; -Bernstein polynomials; modulus of continuity; Voronovskaja type theorem},
language = {eng},
number = {2},
pages = {557-566},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximation properties of bivariate complex $q$-Bernstein polynomials in the case $q>1$},
url = {http://eudml.org/doc/246993},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Mahmudov, Nazim I.
TI - Approximation properties of bivariate complex $q$-Bernstein polynomials in the case $q>1$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 557
EP - 566
AB - In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate $q$-Bernstein polynomials for a function analytic in the polydisc $D_{R_{1}}\times D_{R_{2}}=\lbrace z\in C\colon \vert z\vert <R_{1}\rbrace \times \lbrace z\in C\colon \vert z\vert <R_{1}\rbrace $ for arbitrary fixed $q>1$. We give quantitative Voronovskaja type estimates for the bivariate $q$-Bernstein polynomials for $q>1$. In the univariate case the similar results were obtained by S. Ostrovska: $q$-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.
LA - eng
KW - $q$-Bernstein polynomials; modulus of continuity; Voronovskaja type theorem; -Bernstein polynomials; modulus of continuity; Voronovskaja type theorem
UR - http://eudml.org/doc/246993
ER -

References

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  2. Gal, S. G., Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8, World Scientific New York (2009). (2009) MR2560489
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  6. Ostrovska, S., 10.1007/s10587-008-0079-7, Czech. Math. J. 58 (2008), 1195-1206. (2008) Zbl1174.41010MR2471176DOI10.1007/s10587-008-0079-7
  7. Phillips, G. M., Bernstein polynomials based on the q -integers, Ann. Numer. Math. 4 (1997), 511-518. (1997) Zbl0881.41008MR1422700
  8. Wang, H., Wu, X., 10.1016/j.jmaa.2007.04.014, J. Math. Anal. Appl. 337 (2008), 744-750. (2008) MR2356108DOI10.1016/j.jmaa.2007.04.014
  9. Wu, Z., 10.1016/j.jmaa.2009.04.003, J. Math. Anal. Appl. 357 (2009), 137-141. (2009) Zbl1236.41011MR2526813DOI10.1016/j.jmaa.2009.04.003

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