On Precision Sets of Interpolation Projectors

Abstract

Let (X,d) be a compact metric space and let C(X) denote the real Banach algebra of all continuous functions defined on X. For any continuous projector P on C(X) its precision set prec(P) is defined by

$$ prec(P) = \{ y \in X:\hat yP = \hat y\} $$

where $\hat{y} \in C(X)'$' denotes the point evaluation at y, i. e., the Dirac measure with carrier {y} :

$$ \hat y(f) = f(y)(f \in C(X)). $$