Approximation related to quotient functionals
The best approximation of componentwise positive vectors or positive continuous functions \(f\) is examined by linear combinations \(\hat f = \sum _j \alpha _j \varphi _j\) of given vectors or functions \(\varphi _j\) with respect to functionals \(Q_p, 1\leq p\leq \infty \), involving quotients \(\max\{f/\hat f ,\hat f /f\}\) rather than differences \(|f-\hat f|\). The existence of a best approximating function under mild conditions on \(\{\varphi _j\}_{j=1}^n\) is verified. For discrete data, a best approximating function with respect to \(Q_p\), \(p=1,2,\infty \), is computed by second order cone programming. Special attention is paid to the \(Q_\infty \) functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional \(Q_\infty \) an equivalent characterization of the best approximation by using its extremal set is given. Then this characterization is applied to prove the uniqueness of the best \(Q_\infty \) approximation for Chebyshev sets \(\{\varphi _j\}_{j=1}^n\).
scientific article; zbMATH DE number 5706948