Structure of best radial monotone \(\Phi\)-approximants (Q1916842)

Let \((\Omega, {\mathcal A}, \mu)\) be a finite measure space and let \(\Phi: [0,\infty) \to[0,\infty)\) be a continuous function with \(\Phi(0)=0\) satisfying additional convexity properties. Denote by \({\mathcal L}\) the set of all real valued \({\mathcal A}\)-measurable functions on \(\Omega\). Let \(f\in{\mathcal L}\). The author studies the problem of minimizing the integral \(\int_\Omega \Phi(|f-g |) d\mu\), where \(g\) runs through certain subsets of \({\mathcal L}\). The structure of best approximants is analysed for various situations.