| Title:
|
King type modification of $q$-Bernstein-Schurer operators (English) |
| Author:
|
Ren, Mei-Ying |
| Author:
|
Zeng, Xiao-Ming |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
805-817 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Very recently the $q$-Bernstein-Schurer operators which reproduce only constant function were introduced and studied by C. V. Muraru (2011). Inspired by J. P. King, Positive linear operators which preserve $x^{2}$ (2003), in this paper we modify $q$-Bernstein-Schurer operators to King type modification of $q$-Bernstein-Schurer operators, so that these operators reproduce constant as well as quadratic test functions $x^{2}$ and study the approximation properties of these operators. We establish a convergence theorem of Korovkin type. We also get some estimations for the rate of convergence of these operators by using modulus of continuity. Furthermore, we give a Voronovskaja-type asymptotic formula for these operators. (English) |
| Keyword:
|
King type operator |
| Keyword:
|
$q$-Bernstein-Schurer operator |
| Keyword:
|
Korovich type approximation theorem |
| Keyword:
|
rate of convergence |
| Keyword:
|
Voronovskaja-type result |
| Keyword:
|
modulus of continuity |
| MSC:
|
41A10 |
| MSC:
|
41A25 |
| MSC:
|
41A36 |
| idZBL:
|
Zbl 06282112 |
| idMR:
|
MR3125656 |
| DOI:
|
10.1007/s10587-013-0054-9 |
| . |
| Date available:
|
2013-10-07T12:10:24Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143491 |
| . |
| Reference:
|
[1] Agratini, O., Nowak, G.: On a generalization of Bleimann, Butzer and Hahn operators based on $q$-integers.Math. Comput. Modelling 53 (2011), 699-706. Zbl 1217.33033, MR 2769444, 10.1016/j.mcm.2010.10.006 |
| Reference:
|
[2] Aral, A.: A generalization of Szász-Mirakyan operators based on $q$-integers.Math. Comput. Modelling 47 (2008), 1052-1062. Zbl 1144.41303, MR 2413735, 10.1016/j.mcm.2007.06.018 |
| Reference:
|
[3] Chen, W. Z.: Operators Approximation Theory.Xiamen University Press Xiamen (1989), Chinese. |
| Reference:
|
[4] DeVore, R. A., Lorentz, G. G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften 303.Springer Berlin (1993). MR 1261635, 10.1007/978-3-662-02888-9 |
| Reference:
|
[5] Doğru, O., Örkcü, M.: Statistical approximation by a modification of $q$-Meyer-König Zeller operators.Appl. Math. Lett. 23 (2010), 261-266. MR 2565187, 10.1016/j.aml.2009.09.018 |
| Reference:
|
[6] Gal, S. G.: Voronovskaja's theorem, shape preserving properties and iterations for complex $q$-Bernstein polynomials.Stud. Sci. Math. Hung. 48 (2011), 23-43. MR 2868175 |
| Reference:
|
[7] Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications 34.Cambridge University Press Cambridge (1990). MR 1052153 |
| Reference:
|
[8] Gupta, V., Radu, C.: Statistical approximation properties of $q$-Baskakov-Kantorovich operators.Cent. Eur. J. Math. 7 (2009), 809-818. Zbl 1183.41015, MR 2563452, 10.2478/s11533-009-0055-y |
| Reference:
|
[9] Kac, V., Cheung, P.: Quantum Calculus. Universitext.Springer New York (2002). MR 1865777 |
| Reference:
|
[10] King, J. P.: Positive linear operators which preserve $x^{2}$.Acta Math. Hung. 99 (2003), 203-208. Zbl 1027.41028, MR 1973095, 10.1023/A:1024571126455 |
| Reference:
|
[11] Mahmudov, N. I.: Approximation properties of complex $q$-Szász-Mirakjan operators in compact disks.Comput. Math. Appl. 60 (2010), 1784-1791. Zbl 1202.30061, MR 2679142, 10.1016/j.camwa.2010.07.009 |
| Reference:
|
[12] Mahmudov, N. I.: Approximation by genuine $q$-Bernstein-Durrmeyer polynomials in compact disks.Hacet. J. Math. Stat. 40 (2011), 77-89. Zbl 1230.30020, MR 2663881 |
| Reference:
|
[13] Muraru, C.-V.: Note on $q$-Bernstein-Schurer operators.Stud. Univ. Babeş-Bolyai Math. 56 (2011), 489-495. MR 2843706 |
| Reference:
|
[14] Ostrovska, S.: $q$-Bernstein polynomials of the Cauchy kernel.Appl. Math. Comput. 198 (2008), 261-270. Zbl 1149.41002, MR 2403947, 10.1016/j.amc.2007.08.066 |
| Reference:
|
[15] Phillips, G. M.: Bernstein polynomials based on the $q$-integers.Ann. Numer. Math. 4 (1997), 511-518. Zbl 0881.41008, MR 1422700 |
| Reference:
|
[16] Videnskii, V. S.: On $q$-Bernstein polynomials and related positive linear operators.Problems of Modern Mathematics and Mathematical Education Hertzen readings St.-Petersburg (2004), 118-126 Russian. |
| . |
