A characterization of best \(\varphi\)-approximants with applications to multidimensional isotonic approximation
This paper is a continuation of the previous paper of the same authors [J. Approximation Theory 117, No. 2, 279--300 (2002; Zbl 0557.46020)]. Let \((\Omega,{\mathcal A},\mu)\) be a measurable space. The Musielak-Orlicz norm \(\| .\| _\varphi\) and the Musielak-Orlicz space \({L_{\varphi}=L_{\varphi}(\Omega,{\mathcal A},\mu)}\) are defined as usual [see \textit{J. Musielak}, Orlicz spaces and modular spaces. Lecture Notes in Mathematics. 1034 (1983; Zbl 1012.41025)] on the base of a function \({\varphi: \Omega\times \mathbf{R}\to \text\textbf{R}^+}\) with the following properties: (1) \({\varphi(\cdot,a)}\) is a measurable function for every \({a\in \mathbf{R}}\); (2) \({\varphi(\omega,a)=0}\) iff \({a=0}\); (3) \({\varphi(\omega,\cdot)}\) is an even, nonnull, and convex function. Let \(\mathcal L\subset\mathcal A\) be a \(\sigma\)-lattice. The primary goal is to characterize best \(\varphi\)-approximants of \({f\in L_{\varphi}}\) by elements of the convex cone \({L_\varphi(\mathcal L)}\). The authors show that best \(\varphi\)-approximants to \(f\) from \({L_\varphi(\mathcal L)}\) are best \(\varphi\)-approximants to \(f\) from \(L_{\varphi}(\mathcal A_f)\), where the \(\sigma\)-algebra \(\mathcal A_f\) depends only on \(f\). This result transforms a best approximation problem from a convex set into a best approximation problem from a subspace. Applications of this result to the approximation by nondecreasing functions are considered. The second goal of the authors is the to unify and to extend some known uniqueness results. When \(\Omega\subset\mathbf R^n\) and \(L_\varphi(\mathcal L)\) is the set of monotone functions in several variables, the authors give sufficient conditions on the geometry of \(\Omega\) to obtain a uniqueness theorem. A coincidence relation between a function and its best \(\varphi\)-approximant are obtained. The main results are new, even in the classical Lebesgue spaces \(L_p\).
scientific article; zbMATH DE number 2157510