Approximation related to quotient functionals

We examine the best approximation of componentwise positive vectors or positive continuous functions ff by linear combinations fˆ=∑jαjφj of given vectors or functions φjφj with respect to functionals QpQp, 1≤p≤∞1≤p≤∞, involving quotients max{f/fˆ,fˆ/f} rather than differences |f−fˆ|. We verify the existence of a best approximating function under mild conditions on {φj}j=1n. For discrete data, we compute a best approximating function with respect to QpQp, p=1,2,∞p=1,2,∞ by second order cone programming. Special attention is paid to the Q∞Q∞ functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional Q∞Q∞ we give an equivalent characterization of the best approximation by using its extremal set. Then we apply this characterization to prove the uniqueness of the best Q∞Q∞ approximation for Chebyshev sets {φj}j=1n.