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.
José A. Adell. A. Lekuona. "Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus." Abstr. Appl. Anal. 2015 1 - 11, 2015. https://doi.org/10.1155/2015/915358