Approximation related to quotient functionals

Abstract

We examine the best approximation of componentwise positive vectors or positive continuous functions f by linear combinations f ˆ = ∑ j α j φ j of given vectors or functions φ j with respect to functionals Q p , 1 ≤ p ≤ ∞ , involving quotients max { f / f ˆ , f ˆ / f } rather than differences | f − f ˆ | . We verify the existence of a best approximating function under mild conditions on { φ j } j = 1 n . For discrete data, we compute a best approximating function with respect to Q p , p = 1 , 2 , ∞ by second order cone programming. Special attention is paid to the Q ∞ functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional 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 ∞ approximation for Chebyshev sets { φ j } j = 1 n .