Geodesic
Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 783-799 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We introduce and study a one-parameter class of positive linear operators constituting a link between the well-known operators of S. N. Bernstein and their genuine Bernstein-Durrmeyer variants. Several limiting cases are considered including one relating our operators to mappings investigated earlier by Mache and Zhou. A recursion formula for the moments is proved and estimates for simultaneous approximation of derivatives are given.
We introduce and study a one-parameter class of positive linear operators constituting a link between the well-known operators of S. N. Bernstein and their genuine Bernstein-Durrmeyer variants. Several limiting cases are considered including one relating our operators to mappings investigated earlier by Mache and Zhou. A recursion formula for the moments is proved and estimates for simultaneous approximation of derivatives are given.
Classification : 41A10, 41A17, 41A25, 41A28, 41A36, 41A60
Keywords: positive linear operator; Bernstein-type operator; genuine Bernstein-Durrmeyer operator; simultaneous approximation; degree of approximation; moduli of continuity 
@article{CMJ_2010_60_3_a12,
     author = {Gonska, Heiner and P\u{a}lt\u{a}nea, Radu},
     title = {Simultaneous approximation by a class of {Bernstein-Durrmeyer} operators preserving linear functions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {783--799},
     year = {2010},
     volume = {60},
     number = {3},
     mrnumber = {2672415},
     zbl = {1224.41016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a12/}
}
TY  - JOUR
AU  - Gonska, Heiner
AU  - Păltănea, Radu
TI  - Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions
JO  - Czechoslovak Mathematical Journal
PY  - 2010
SP  - 783
EP  - 799
VL  - 60
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a12/
LA  - en
ID  - CMJ_2010_60_3_a12
ER  - 
%0 Journal Article
%A Gonska, Heiner
%A Păltănea, Radu
%T Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions
%J Czechoslovak Mathematical Journal
%D 2010
%P 783-799
%V 60
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a12/
%G en
%F CMJ_2010_60_3_a12
Gonska, Heiner; Păltănea, Radu. Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 783-799. http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a12/

[1] Berens, H., Xu, Y.: On Bernstein-Durrmeyer polynomials with Jacobi weights. In: Approximation Theory and Functional Analysis Academic Press Boston (1991), 25-46. | MR | Zbl

[2] Chen, W.: On the modified Bernstein-Durrmeyer operator. Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China (1987).

[3] Derriennic, M. M.: Sur l'approximation de fonctions intégrables sur $[0,1]$ par des polynômes de Bernstein modifiés. J. Approx. Theory 31 (1981), 325-343 French. | DOI | MR

[4] Durrmeyer, J. L.: Une formule d'inversion de la transformée de Laplace: Applications à la théorie des moments. Thèse de 3ème cycle. Faculté des Sciences Univ. Paris (1967).

[5] Gavrea, I.: The approximation of the continuous functions by means of some positive operators. Result. Math. 30 (1996), 55-66. | DOI | MR

[6] Gonska, H.: Quantitative Korovkin-type theorems on simultaneous approximation. Math. Z. 186 (1984), 419-433. | DOI | MR | Zbl

[7] Gonska, H. H., Kacsó, D., Raşa, I.: On genuine Bernstein-Durrmeyer operators. Result. Math. 50 (2007), 213-225. | DOI | MR

[8] Goodman, T. N. T., Sharma, A.: A modified Bernstein-Schoenberg operator. Proc. Conf. Constructive Theory of Functions, Varna 1987 Bl. Sendov et al. Publ. House Bulg. Acad. Sci. Sofia (1988), 166-173. | MR | Zbl

[9] Kacsó, D.: Certain Bernstein-Durrmeyer type operators preserving linear functions. Habilitationschrift Universität Duisburg-Essen (2006).

[10] Lupaş, A.: Die Folge der Betaoperatoren. Dissertation Universität Stuttgart (1972).

[11] Mache, D. H., Zhou, D. X.: Characterization theorems for the approximation by a family of operators. J. Approx. Theory 84 (1996), 145-161. | DOI | MR | Zbl

[12] Mond, B.: On the degree of approximation by linear positive operators. J. Approx. Theory 18 (1976), 304-306. | DOI | MR | Zbl

[13] Parvanov, P. E., Popov, B. D.: The limit case of Bernstein's operators with Jacobi weights. Math. Balk. (N.S.) 8 (1994), 165-177. | MR | Zbl

[14] Pǎltǎnea, R.: Sur un operateur polynômial défini sur l'ensemble des fonctions intégrables. Babeş Bolyai Univ., Fac. Math., Res. Semin. 2 (1983), 101-106 French. | MR

[15] Pǎltǎnea, R.: Une propriété d'extrémalité des valeurs propres des opérateurs polynômiaux de Durrmeyer généralisés. Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 15 (1986), 57-64 French. | MR

[16] Pǎltǎnea, R.: Approximation Theory Using Positive Linear Operators. Birkhäuser Boston (2004). | MR

[17] Pǎltǎnea, R.: A class of Durrmeyer type operators preserving linear functions. Annals of the Tiberiu Popoviciu Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca), vol. 5 (2007), 109-118.

[18] Sauer, T.: The genuine Bernstein-Durrmeyer operator on a simplex. Result. Math. 26 (1994), 99-130. | DOI | MR

[19] Waldron, S.: A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights. J. Approx. Theory 122 (2003), 141-150. | DOI | MR | Zbl