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Keywords:
$q$-Bernstein polynomials; modulus of continuity; Voronovskaja type theorem
Summary:
In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate $q$-Bernstein polynomials for a function analytic in the polydisc $D_{R_{1}}\times D_{R_{2}}=\{z\in C\colon \vert z\vert <R_{1}\} \times \{ z\in C\colon \vert z\vert <R_{1}\}$ 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.
References:
[1] Butzer, P. L.: On two-dimensional Bernstein polynomials. Can. J. Math. 5 (1953), 107-113. DOI 10.4153/CJM-1953-014-2 | MR 0052573 | Zbl 0050.07002
[2] Gal, S. G.: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific New York (2009). MR 2560489
[3] Hildebrandt, T. H., Schoenberg, I. J.: On linear functional operations and the moment problem for a finite interval in one or several dimensions. Ann. Math. 34 (1933), 317-328. DOI 10.2307/1968205 | MR 1503109 | Zbl 0006.40204
[4] Mahmudov, N. I.: Korovkin-type theorems and applications. Cent. Eur. J. Math. 7 (2009), 348-356. DOI 10.2478/s11533-009-0006-7 | MR 2506971 | Zbl 1179.41024
[5] Ostrovska, S.: $q$-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232-255. DOI 10.1016/S0021-9045(03)00104-7 | MR 1990098 | Zbl 1093.41013
[6] Ostrovska, S.: The sharpness of convergence results for $q$-Bernstein polynomials in the case $q>1$. Czech. Math. J. 58 (2008), 1195-1206. DOI 10.1007/s10587-008-0079-7 | MR 2471176 | Zbl 1174.41010
[7] Phillips, G. M.: Bernstein polynomials based on the $q$-integers. Ann. Numer. Math. 4 (1997), 511-518. MR 1422700 | Zbl 0881.41008
[8] Wang, H., Wu, X.: Saturation of convergence for $q$-Bernstein polynomials in the case $q>1$. J. Math. Anal. Appl. 337 (2008), 744-750. DOI 10.1016/j.jmaa.2007.04.014 | MR 2356108
[9] Wu, Z.: The saturation of convergence on the interval $[0;1]$ for the $q$-Bernstein polynomials in the case $q>1$. J. Math. Anal. Appl. 357 (2009), 137-141. DOI 10.1016/j.jmaa.2009.04.003 | MR 2526813 | Zbl 1236.41011

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