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Abstract and Applied Analysis
Volume 2015, Article ID 915358, 11 pages
http://dx.doi.org/10.1155/2015/915358
Research Article

Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus

Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received 16 February 2015; Accepted 24 April 2015

Academic Editor: Milan Pokorny

Copyright © 2015 José A. Adell and A. Lekuona. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give accurate estimates of the constants appearing in direct inequalities of the form   , , ,  and   where is a positive linear operator reproducing linear functions and acting on real functions defined on the interval , is a certain subset of such functions, is the usual second modulus of , and is an appropriate weight function. We show that the size of the constants mainly depends on the degree of smoothness of the functions in the set and on the distance from the point to the boundary of . We give a closed form expression for the best constant when is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.