On approximation by rational functions in Musielak-Orlicz spaces (Q6632943)

The authors extend the results from [\textit{W. M. Kozlowski}, J. Approx. Theory 264, Article ID 105535, 14 p. (2021; Zbl 1462.41015); Appl. Set-Valued Anal. Optim. 4, No. 3, 337--348 (2022; \url{doi:10.23952/asvao.4.2022.3.07})] to the problem of best approximation by rational functions in a larger class of Musielak-Orlicz spaces of real-valued measurable functions over the unit interval, equipped with the Lebesgue measure. The theory of Musielak-Orlicz spaces is briefly presented with the intent of equipping the reader with all necessary information. As a result, the paper is self-contained from the Musielak-Orlicz space background perspective. The authors use an adaptation of the concept of approximatively compact sets, frequently used when dealing with approximation in normed spaces. They do not assume any form of uniform convexity and consequently, in general, they are not in a position to get uniqueness of the best approximants. Hence, the nonlinear projection operator must be understood as multi-valued.\N\NTheorem 3.4 is the main result of the paper. In its essence this theorem says that, under some very general assumptions, every element \(f\) of the Musielak-Orlicz space \(L^\phi\) can be approximated by a rational function, and that the set of all best approximants of \(f\) is compact in some sense. Moreover, the multi-valued projection operator is upper semi-continuous in some standard way.\N\NThe final section shows how the results of the preceding sections can be applied to the approximation problems in weighted variable exponent Lebesgue spaces, as well as in classical Orlicz spaces \(L^\phi\) where the Musielak-Orlicz function \(\phi(t, u)\) does not directly depend on the first variable.\N\NApplications to image processing in topography and biomedical fields are presented.