On modular approximants in sequential convergence spaces

Abstract

Let Xρ be a modulated convergence space, that is, a modular space equipped with a sequential convergence structure. Given an element x of Xρ, we consider the minimisation problem of finding x0∈C such that ρ(x−xo)=inf{ρ(x−y):y∈C}, where ρ is a convex modular and C is a closed convex subset of Xρ. Such an element x0 is called a best approximant. We prove existence and uniqueness of such a best approximant in a large classes of modulated convergence spaces, provided ρ is uniformly convex. We also touch upon an interesting subject of semicontinuity of the related modular projection. 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 nonlinear prediction theory.