Rates of convergence of Chlodovsky-Kantorovich polynomials in classes of locally integrable functions
Paulina Pych-Taberska
Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 29 (2009), p. 53-66 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

In this paper we establish an estimation for the rate of pointwise convergence of the Chlodovsky-Kantorovich polynomials for functions f locally integrable on the interval [0,∞). In particular, corresponding estimation for functions f measurable and locally bounded on [0,∞) is presented, too.

Publié le : 2009-01-01
EUDML-ID : urn:eudml:doc:271163 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1104,
     author = {Paulina Pych-Taberska},
     title = {Rates of convergence of Chlodovsky-Kantorovich polynomials in classes of locally integrable functions},
     journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
     volume = {29},
     year = {2009},
     pages = {53-66},
     zbl = {1195.41017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1104}
}

Paulina Pych-Taberska. Rates of convergence of Chlodovsky-Kantorovich polynomials in classes of locally integrable functions. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 29 (2009) pp. 53-66. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1104/

[000] [1] J. Albrycht and J. Radecki, On a generalization of the theorem of Voronovskaya, Zeszyty Naukowe UAM, Zeszyt 2, Poznań (1960), 1-7. | Zbl 0094.03704

[001] [2] R. Bojanic and O. Shisha, Degree of L₁ approximation to integrable functions by modified Bernstein polynomials, J. Approx. Theory 13 (1975), 66-72. | Zbl 0305.41009

[002] [3] P.L. Butzer and H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein-Chlodovsky polynomials and the Szász-Mirakyan operator, Comment. Math., to appear. | Zbl 1236.41022

[003] [4] Z.A. Chanturiya, Modulus of variation of functions and its application in the theory of Fourier series, Dokl. Akad. Nauk SSSR 214 (1974), 63-66.

[004] [5] I. Chlodovsky, Sur le développement des fonctions définies dans un intervalle infini en séries de polynomes de M.S. Bernstein, Compositio Math. 4 (1937), 380-393. | Zbl 63.0237.01

[005] [6] M. Heilmann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl. 5 (1) (1989), 105-127. | Zbl 0669.41014

[006] [7] H. Karsli and E. Ibikli, Rate of convergence of Chlodovsky type Bernstein operators for functions of bounded variation, Numer. Funct. Anal. Optim. 28 (3-4) (2007), 367-378. | Zbl 1117.41020

[007] [8] G.G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953.

[008] [9] P. Pych-Taberska, Some properties of the Bézier-Kantorovich type operators, J. Approx. Theory 123 (2003), 256-269.

[009] [10] L.C. Young, General inequalities for Stieltjes integrals and the convergence of Fourier series, Math. Annalen 115 (1938), 581-612. | Zbl 64.0198.03

[010] [11] X.M. Zeng, Bounds for Bernstein basis functions and Meyer-König and Zeller basis functions, J. Math. Anal. Appl. 219 (2) (1998), 364-376. | Zbl 0909.41015

[011] [12] X.M. Zeng and A. Piriou On the rate of convergence of two Berstein-Bézier type operators for bounded variation functions, J. Approx. Theory 95 (1998), 369-387.