Lipschitz continuity and Gateaux differentiability 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 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 and has a Gateaux derivative on a dense set of functions in C X, R k .

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

“…The following theorem [4] shows that there is a particular set of functions in C X, R k at which B, by the result of Cheney, has Lipschitz continuity of order 1.…”

“…Part of the original motivation for this paper comes from the well-known [12] fact that in Hilbert space even though the projection operator onto a closed subspace (the best approximation operator associated with that subspace) has strong unicity of order 2, but not of order 1 in general, it is Lipschitz continuous of order 1. Bartelt and Swetits [4] showed that the best approximation operator is Lipschitz continuous of order 1 on a dense subset of C(X, R k ) when X is finite and G is a Haar subspace of C(X, R k ) and they conjectured that the best approximation operator is globally Lipschitz continuous of order 1 in this case. They verified the conjecture in the case k = 2 when G is the two dimensional subspace of constant vectors.…”