Geodesic
Approximation properties of bivariate complex $q$-Bernstein polynomials in the case $q>1$
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 557-566

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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}}\times D_{R_{2}}=\{z\in C\colon \vert z\vert $ 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.
DOI : 10.1007/s10587-012-0029-2
Classification : 33D15, 41A10, 41A35
Keywords: $q$-Bernstein polynomials; modulus of continuity; Voronovskaja type theorem 
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Mahmudov, Nazim I. Approximation properties of bivariate complex $q$-Bernstein polynomials in the case $q>1$. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 557-566. doi: 10.1007/s10587-012-0029-2

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