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

Abstract

We give accurate estimates of the constants $C_n(\mathcal{A}(I),x)$ appearing in direct inequalities of the form $|L_{n}f(x)-f(x)|\le C_n(\mathcal{A}(I),x){\omega }_2$ $\left(f;\sigma (x)/\sqrt{n}\right)$, $f\in \mathcal{A}(I)$, $x\in I$, and $n=\mathrm{1,2},\dots ,$ where $L_n$ is a positive linear operator reproducing linear functions and acting on real functions $f$ defined on the interval $I$, $\mathcal{A}(I)$ is a certain subset of such functions, ${\omega }_2(f;·)$ is the usual second modulus of $f$, and $\sigma (x)$ is an appropriate weight function. We show that the size of the constants $C_n(\mathcal{A}(I),x)$ mainly depends on the degree of smoothness of the functions in the set $\mathcal{A}(I)$ and on the distance from the point $x$ to the boundary of $I$. We give a closed form expression for the best constant when $\mathcal{A}(I)$ is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.