Journal of Approximation Theory
When G is a finite dimensional Haar subspace of C(X, Rk), 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, Rk) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (H˝older) condition of order 1/2. This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.
Original Publication Citation
Bartelt, M., & Swetits, J. (2008). Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation. Journal of Approximation Theory, 152(2), 161-166. doi:10.1016/j.jat.2007.12.002
Bartelt, Martin and Swetits, John, "Lipschitz Continuity of the Best Approximation Operator in Vector-Valued Chebyshev Approximation" (2008). Mathematics & Statistics Faculty Publications. 55.