Rate of convergence of positive linear operators using an extended complete Tchebycheff system (Q910989)

Let [a,b] be a compact interval and let \(\{L_ j\}\) be a sequence of positive linear operators from \(C^{n+1}[a,b]\) to C[a,b]. The convergence of \(L_ j\) to the unit operator I is closely related to the weak convergence of a sequence of positive finite measures \(\mu_ j\) to the unit measure \(\delta_ t\) where \(t\in [a,b]\). The author obtains very general estimates with rates for the error \(| \int_{[a,b]}f d\mu_ j-f(t)|\) where \(f\in C^{n+1}[a,b]\), in the presence of an extended complete Chebyshev system. These estimates lead to sharp or nearly sharp inequalities and are related to the theory of best \(L_ 1\) approximations by generalized polynomials. For earlier work in this area, see \textit{O. Shisha} and \textit{B. Mond} [Proc. Nat. Acad. Sci. USA 60, 1196-1200 (1968; Zbl 0164.071)].