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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $D_{R_{1}}\times D_{R_{2}}=\{z\in C\colon \vert z\vert 1$. 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.
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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