Rates of convergence of Chlodovsky-Kantorovich polynomials in classes of locally integrable functions

Paulina Pych-Taberska

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2009)

  • Volume: 29, Issue: 1, page 53-66
  • ISSN: 1509-9407

Abstract

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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.

How to cite

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Paulina Pych-Taberska. "Rates of convergence of Chlodovsky-Kantorovich polynomials in classes of locally integrable functions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29.1 (2009): 53-66. <http://eudml.org/doc/271163>.

@article{PaulinaPych2009,
abstract = {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.},
author = {Paulina Pych-Taberska},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {Chlodovsky polynomial; Kantorovich polynomial; rate of convergence},
language = {eng},
number = {1},
pages = {53-66},
title = {Rates of convergence of Chlodovsky-Kantorovich polynomials in classes of locally integrable functions},
url = {http://eudml.org/doc/271163},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Paulina Pych-Taberska
TI - Rates of convergence of Chlodovsky-Kantorovich polynomials in classes of locally integrable functions
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2009
VL - 29
IS - 1
SP - 53
EP - 66
AB - 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.
LA - eng
KW - Chlodovsky polynomial; Kantorovich polynomial; rate of convergence
UR - http://eudml.org/doc/271163
ER -

References

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  1. [1] J. Albrycht and J. Radecki, On a generalization of the theorem of Voronovskaya, Zeszyty Naukowe UAM, Zeszyt 2, Poznań (1960), 1-7. Zbl0094.03704
  2. [2] R. Bojanic and O. Shisha, Degree of L₁ approximation to integrable functions by modified Bernstein polynomials, J. Approx. Theory 13 (1975), 66-72. Zbl0305.41009
  3. [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. Zbl1236.41022
  4. [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. 
  5. [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. Zbl63.0237.01
  6. [6] M. Heilmann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl. 5 (1) (1989), 105-127. Zbl0669.41014
  7. [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. Zbl1117.41020
  8. [8] G.G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953. 
  9. [9] P. Pych-Taberska, Some properties of the Bézier-Kantorovich type operators, J. Approx. Theory 123 (2003), 256-269. 
  10. [10] L.C. Young, General inequalities for Stieltjes integrals and the convergence of Fourier series, Math. Annalen 115 (1938), 581-612. Zbl64.0198.03
  11. [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. Zbl0909.41015
  12. [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. 

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