| Title:
|
Approximation and shape preserving properties of the nonlinear Bleimann-Butzer-Hahn operators of max-product kind (English) |
| Author:
|
Bede, Barnabás |
| Author:
|
Coroianu, Lucian |
| Author:
|
Gal, Sorin G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2010 |
| Pages:
|
397-415 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Starting from the study of the Shepard nonlinear operator of max-prod type in (Bede, Nobuhara et al., 2006, 2008), in the book (Gal, 2008), Open Problem 5.5.4, pp. 324--326, the Bleimann-Butzer-Hahn max-prod type operator is introduced and the question of the approximation order by this operator is raised. In this paper firstly we obtain an upper estimate of the approximation error of the form $\omega_{1}(f;(1+x)^{\frac{3}{2}}\sqrt{x/n})$. A consequence of this result is that for each compact subinterval $[0,a]$, with arbitrary $a>0$, the order of uniform approximation by the Bleimann-Butzer-Hahn operator is less than ${\mathcal{O}}(1/\sqrt{n})$. Then, one proves by a counterexample that in a sense, for arbitrary $f$ this order of uniform approximation cannot be improved. Also, for some subclasses of functions, including for example the bounded, nondecreasing concave functions, the essentially better order $\omega_{1}(f;(x+1)^{2}/n)$ is obtained. Shape preserving properties are also investigated. (English) |
| Keyword:
|
nonlinear Bleimann-Butzer-Hahn operator of max-product kind |
| Keyword:
|
degree of approximation |
| Keyword:
|
shape preserving properties |
| MSC:
|
41A25 |
| MSC:
|
41A29 |
| MSC:
|
41A30 |
| idZBL:
|
Zbl 1224.41060 |
| idMR:
|
MR2741873 |
| . |
| Date available:
|
2010-09-02T14:12:42Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140716 |
| . |
| Reference:
|
[1] Bede B., Gal S.G.: Approximation by nonlinear Bernstein and Favard-Szász-Mirakjan operators of max-product kind.J. Concrete and Applicable Mathematics 8 (2010), no. 2, 193–207. MR 2606257 |
| Reference:
|
[2] Bede B., Coroianu L., Gal S.G.: Approximation and shape preserving properties of the Bernstein operator of max-product kind.Int. J. Math. Math. Sci. 2009, Art. ID 590589, 26 pp., doi:10.1155/2009/590589. Zbl 1188.41016, MR 2570725 |
| Reference:
|
[3] Bede B., Coroianu L., Gal S.G.: Approximation by truncated nonlinear Favard-Szász-Mirakjan operators of max-product kind.Demonstratio Math.(to appear). MR 2796766 |
| Reference:
|
[4] Bede B., Nobuhara H., Fodor J., Hirota K.: Max-product Shepard approximation operators.J. Advanced Computational Intelligence and Intelligent Informatics 10 (2006), 494–497. |
| Reference:
|
[5] Bede B., Nobuhara H., Daňková M., Di Nola A.: Approximation by pseudo-linear operators.Fuzzy Sets and Systems 159 (2008), 804–820. MR 2403975 |
| Reference:
|
[6] Bleimann G., Butzer P.L., Hahn L.: A Bernstein-type operator approximating continuous functions on the semi-axis.Indag. Math. 42 (1980), 255–262. Zbl 0437.41021, MR 0587054 |
| Reference:
|
[7] Duman O.: Statistical convergence of max-product approximating operators.Turkish J. Math. 33 (2009), 1–14. MR 2721963 |
| Reference:
|
[8] Gal S.G.: Shape-Preserving Approximation by Real and Complex Polynomials.Birkhäuser, Boston-Basel-Berlin, 2008. Zbl 1154.41002, MR 2444986 |
| Reference:
|
[9] Khan R.A.: A note on a Bernstein-type operator of Bleimann, Butzer and Hahn.J. Approx. Theory 53 (1988), no. 3, 295–303. Zbl 0676.41024, MR 0947433, 10.1016/0021-9045(88)90024-X |
| Reference:
|
[10] Popoviciu T.: Deux remarques sur les fonctions convexes.Bull. Soc. Sci. Acad. Roumaine 220 (1938), 45–49. Zbl 0021.11605 |
| . |
