A generalization of Stancu-Mühlbach operators (Q2644880)

Let X be a metrizable convex compact subset embedded in a local convex Hausdorff space and \((a_ n)_{n\in {\mathbb{N}}}\) a sequence of nonnegative real numbers converging to zero. The well-known sequence \((Q_ n,a_ n)_{n\in {\mathbb{N}}}\) of Stancu-Mühlbach operators on C[0,1] (reducing for \(a_ n=0\) and \(a_ n=-1/n\) respectively to Bernstein and Lagrange interpolation operators respectively is generalized to operators acting on the space C(X,\({\mathbb{R}})\). These operators are associated with a nonnegative projection T of C(X,\({\mathbb{R}})\) and an infinite lower triangular stochastic matrix P. The main part of the paper studies the limit behaviour of this sequence of operators and of its iterates. Generalizing a modulus of continuity which has been introduced by Schnabl and Fehlbecker independently the author finally obtains a quantitative result.