Rate of convergence of positive linear operators using an extended complete Tchebycheff system


Let [a,b] ⊂ Rand let {Lj}j ε{lunate} N be a sequence of positive linear operators from Cn + 1([a, b]) to C([a, b]), n ≥ 0. The convergence of Lj to the unit operator I is closely related to the weak convergence of a sequence of positive finite measures μj to the unit measure δt, t ε{lunate} [a, b]. Very general estimates with rates are given for the error |∝[a, b] f dμj - f(t)|, where f ε{lunate} Cn + 1([a, b]), in the presence of an extended complete Tchebycheff system. These lead to sharp or nearly sharp inequalities of Shisha-Mond type and are connected to the theory of best L1 approximations by generalized polynomials. © 1989.

Publication Title

Journal of Approximation Theory