Stechkin-Marchaud Type Inequalities in L_p for Linear Combination of Bernstein-Durrmeyer Operators

Feng Guo, Meiqin Ke

Library, Taizhou University, Zhejiang,Taizhou 317000, China.
School of Mathematics and Information Engineering, Taizhou University, Zhejiang,Taizhou 317000, China.

DOI: 10.4236/ojapps.2012.24B052                

Abstract

In this paper, we use the equivalence relation between K-functional and modulus of smoothness, and give the Stechkin-Marchaud-type inequalities for linear combination of Bernstein-Durrmeyer operators . Moreover, we obtain the inverse result of approximation for linear combination of Bernstein-Durrmeyer operators with . Meanwhile we unify and extend some previous results.

Keywords

Bernstein-Durrmeyer operators; linear combination; K-functional;Stechkin-Marchaud-type inequalities; modulus of smoothness

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M..M..Derrieinnic, “On Multivariate Approximation by Bernstein-type Polynomials,” J. Approx. Theory, vol.45, pp. 155-156, 1985.
[2]	Z. Ditzian, K. G. Ivanov “ Bernstein-type operators and their derivatives,” J Approx. Theory, vol.56, pp.72-90, 1989 .
[3]	D.X.Zhou, “On smoothness characterized by Bernstein type operators,” J. Approx Theory, vol.81, pp.303-315, 1995.
[4]	S.S. Guo，C.X.Li , “Approxition by Linear Commbinations of Bernstein-Durrmeyer Operators,” Journal of Lanzhou University (Natural Sciences) , vol. 36(6), pp. 13-16, 2000.
[5]	Z.Ditzian, V.Totik, “ Moduli of Smoothness,”Springer-Verlag, New York, 1987.
[6]	V. Wiekeren, weakptype inequalities for Kantorovich polynomials and related operators. Indng Math., 1987, 90(1).111-120