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

Abstract EN

We give accurate estimates of the constants Cn(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,…, where Ln is a positive linear operator reproducing linear functions and acting on real functions f defined on the interval I, A(I) is a certain subset of such functions, ω2(f;·) is the usual second modulus of f, and σ(x) is an appropriate weight function.

We show that the size of the constants Cn(A(I),x) mainly depends on the degree of smoothness of the functions in the set 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 A(I) is a certain set of continuous piecewise linear functions.

As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.