| Title:
|
Approximation properties of bivariate complex $q$-Bernstein polynomials in the case $q>1$ (English) |
| Author:
|
Mahmudov, Nazim I. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
557-566 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
$q$-Bernstein polynomials |
| Keyword:
|
modulus of continuity |
| Keyword:
|
Voronovskaja type theorem |
| MSC:
|
33D15 |
| MSC:
|
41A10 |
| MSC:
|
41A35 |
| idZBL:
|
Zbl 1265.33036 |
| idMR:
|
MR2990194 |
| DOI:
|
10.1007/s10587-012-0029-2 |
| . |
| Date available:
|
2012-06-08T09:53:13Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142846 |
| . |
| Reference:
|
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| Reference:
|
[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 |
| Reference:
|
[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. Zbl 0006.40204, MR 1503109, 10.2307/1968205 |
| Reference:
|
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| Reference:
|
[5] Ostrovska, S.: $q$-Bernstein polynomials and their iterates.J. Approximation Theory 123 (2003), 232-255. Zbl 1093.41013, MR 1990098, 10.1016/S0021-9045(03)00104-7 |
| Reference:
|
[6] Ostrovska, S.: The sharpness of convergence results for $q$-Bernstein polynomials in the case $q>1$.Czech. Math. J. 58 (2008), 1195-1206. Zbl 1174.41010, MR 2471176, 10.1007/s10587-008-0079-7 |
| Reference:
|
[7] Phillips, G. M.: Bernstein polynomials based on the $q$-integers.Ann. Numer. Math. 4 (1997), 511-518. Zbl 0881.41008, MR 1422700 |
| Reference:
|
[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. MR 2356108, 10.1016/j.jmaa.2007.04.014 |
| Reference:
|
[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. Zbl 1236.41011, MR 2526813, 10.1016/j.jmaa.2009.04.003 |
| . |
