Journal of Approximation Theory
When G is a finite-dimensional Haar subspace of C ( X, Rk), the vector-valued functions (including complex-valued functions when k is 2) frorn 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 (Holder) condition of order 1/2. This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C ( X, Rk).
Original Publication Citation
Bartelt, M., & Swetits, J. (2007). Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation. Journal of Approximation Theory, 148(2), 177-193. doi:10.1016/j.jat.2007.03.005
Bartelt, Martin and Swetits, John, "Lipschitz Continuity and Gateaux Differentiability of the Best Approximation Operator in Vector-Valued Chebyshev Approximation" (2007). Mathematics & Statistics Faculty Publications. 54.