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On approximation in spaces of continuous functions

This paper deals with approximation of certain operators defined on the space C(X) of real-valued continuous functions on an arbitrary compact metric space (X, d). In particular the problem of giving quantitative Korovkin type theorems for approximation by positive linear operators is solved. This is achieved by using a smoothing approach and the least concave majorant of the modulus of continuity of a function f in C(X). Several new estimates are given as applications, including such for Shepard's method of metric interpolation.