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

Abstract

We give accurate estimates of the constantsCn(A(I),x)appearing in direct inequalities of the form|Lnf(x)-f(x)|≤Cn(A(I),x)ω2  f;σ(x)/n,f∈A(I),x∈I,  and  n=1,2,…,whereLnis a positive linear operator reproducing linear functions and acting on real functionsfdefined on the intervalI,A(I)is a certain subset of such functions,ω2(f;·)is the usual second modulus off, andσ(x)is an appropriate weight function. We show that the size of the constantsCn(A(I),x)mainly depends on the degree of smoothness of the functions in the setA(I)and on the distance from the pointxto the boundary ofI. We give a closed form expression for the best constant whenA(I)is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.