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

摘要

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 (Holder) condition of order 1/2. This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1. (c) 2008 Elsevier Inc. All rights reserved.