On modular approximants in sequential convergence spaces

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) is an element of C such that rho(x - x(0)) = inf{rho(x - y) : y is an element of C}, where p is a convex modular and C is a closed convex subset of X-rho. Such an element x(0) is called a best approximant. We prove existence and uniqueness of such a best approximant in a large classes of modulated convergence spaces, provided rho 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. (C) 2021 Elsevier Inc. All rights reserved.