Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

Abstract

When G is a finite dimensional Haar subspace of C ( X , R k ) , the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C ( X , R k ) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order 1 2 . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.