Interpolation and approximation from convex sets. II. Infinite-dimensional interpolation

Let X and Y be topological vector spaces, A be a continuous linear map from X to Y, C⊂X, B be a convex set dense in C, and d∈Y be a data point. Conditions are derived guaranteeing the set B∩A−1(d) to be nonempty and dense in C∩A−1(d). The paper generalizes earlier results by the authors to the case where Y is infinite dimensional. The theory is illustrated with two examples concerning the existence of smooth monotone extensions of functions defined on a domain of the Euclidean space to a larger domain.