Wojciech M. Kozlowski, Notes on modular projections

Full Text: PDF
DOI: 10.23952/asvao.4.2022.3.07
Volume 4, Issue 3, 1 December 2022, Pages 337-348

Abstract. Let $X_{\rho}$ be a modulated convergence space, that is, a modular space equipped with a sequential convergence structure. Given an element $x$ of $X_{\rho}$, we consider the minimisation problem of finding $x_0 \in K$ such that $\rho(x-x_o) = \inf \{\rho(x - y): y \in K \}$, where $\rho$ is a modular, and $K$ is a subset of $X_{\rho}$. Such an element $x_0$ is called a best approximant. We prove an existence of best approximants in a large class of modulated convergence spaces. We also investigate semicontinuity of the related modular projection which in general is a nonlinear multi-valued operator. Problems of finding best approximants are important in approximation theory and probability theory. In particular, we show how our results can be applied to the approximation of functions in variable Lebesgue spaces by rational functions.