A study of positive linear operators by the method of moments, one-dimensional case


Let [a, b] ⊂R and let N be a sequence of positive linear operators from Cn[a, b] (nε{lunate}Z+) to C[a, b]. The convergence of Lj to the identity operator I is closely related to the weak convergence of a sequence of finite measure μj, to the unit (Dirac) measure δx0, x0 ε{lunate} [a, b]. New estimates are given for the remainder |∝[a,b]f{hook}dμj - f{hook}(x0)|, where f{hook} ε{lunate} Cn([a, b]). Using moment methods, Shisha-Mond-type best or nearly best upper bounds are established for various choices of [a, b], n and given moments of μj. Some of them lead to attainable inequalities. The optimal functions/measures are typically spline functions and finitely supported measures. The corresponding inequalities involve the first modulus of continuity of f{hook}(n) (the nth derivative of f{hook}) or a modification of it. Several applications of these results are given. © 1985.

Publication Title

Journal of Approximation Theory