| Title:
|
On approximation of functions by certain operators preserving $x^2$ (English) |
| Author:
|
Rempulska, Lucyna |
| Author:
|
Tomczak, Karolina |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
579-593 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving $e_k (x)=x^k$, $k=0,2$. Using a modification of certain operators $L_n$ preserving $e_0$ and $e_1$, we introduce operators $L_n^*$ which preserve $e_0$ and $e_2$ and next we define operators $L_{n;r}^{*}$ for $r$-times differentiable functions. We show that $L_n^*$ and $L_{n;r}^{*}$ have better approximation properties than $L_n$ and $L_{n;r}$. (English) |
| Keyword:
|
positive linear operators |
| Keyword:
|
polynomial weighted space |
| Keyword:
|
degree of approximation |
| MSC:
|
41A25 |
| MSC:
|
41A36 |
| idZBL:
|
Zbl 1212.41054 |
| idMR:
|
MR2493939 |
| . |
| Date available:
|
2009-05-05T17:13:16Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119747 |
| . |
| Reference:
|
[1] Baskakov V.A.: An example of a sequence of linear positive operators in the space of continuous functions.Dokl. Akad. Nauk SSSR 113 (1957), 249-251. MR 0094640 |
| Reference:
|
[2] Becker M.: Global approximation theorems for Szász-Mirakyan and Baskakov operators in polynomial weight spaces.Indiana Univ. Math. J. 27 1 (1978), 127-142. MR 0493079, 10.1512/iumj.1978.27.27011 |
| Reference:
|
[3] De Vore R.A.: The Approximation of Continuous Functions by Positive Linear Operators.Springer, Berlin, New York, 1972. MR 0420083 |
| Reference:
|
[4] De Vore R.A., Lorentz G.G.: Constructive Approximation.Springer, Berlin, New York, 1993. MR 1261635 |
| Reference:
|
[5] Ditzian Z., Totik V.: Moduli of Smoothness.Springer, New York, 1987. Zbl 0715.46043, MR 0914149 |
| Reference:
|
[6] Duman O., Özarslan M.A.: MKZ type operators providing a better estimation on $[1/2,1)$.Canad. Math. Bull. 50 (2007), 434-439. Zbl 1132.41318, MR 2344178, 10.4153/CMB-2007-042-8 |
| Reference:
|
[7] Duman O., Özarslan M.A.: Szász-Mirakyan type operators providing a better error estimation.Appl. Math. Lett. 20 12 (2007), 1184-1188. MR 2384243, 10.1016/j.aml.2006.10.007 |
| Reference:
|
[8] King J.P.: Positive linear operators which preserve $x^2$.Acta Math. Hungar. 99 (2003), 203-208. Zbl 1027.41028, MR 1973095, 10.1023/A:1024571126455 |
| Reference:
|
[9] Kirov G.H.: A generalization of the Bernstein polynomials.Math. Balcanica 2 2 (1992), 147-153. Zbl 0838.41017, MR 1182946 |
| Reference:
|
[10] Kirov G.H., Popova L.: A generalization of the linear positive operators.Math. Balcanica 7 2 (1993), 149-162. Zbl 0833.41016, MR 1270375 |
| Reference:
|
[11] Rempulska L., Walczak Z.: Modified Szász-Mirakyan operators.Math. Balcanica 18 (2004), 53-63. Zbl 1079.41022, MR 2076077 |
| Reference:
|
[12] Rempulska L., Skorupka M.: Approximation properties of modified gamma operators.Integral Transforms Spec. Funct. 18 9-10 (2007), 653-662. Zbl 1148.41025, MR 2356794, 10.1080/10652460701510527 |
| Reference:
|
[13] Rempulska L., Skorupka M.: On approximation by Post-Widder and Stancu operators preserving $x^2$.Kyung. Math. J., to appear. MR 2527373 |
| Reference:
|
[14] Stancu D.D.: On the beta approximating operators of second kind.Rev. Anal. Numér. Théor. Approx. 24 (1-2) (1995), 231-239. Zbl 0856.41019, MR 1608424 |
| . |
