# Best approximants in modular function spaces

## Abstract

In this paper we consider the existence of best approximants in modular function spaces by elements of sublattices. Modular function spaces are the natural generalization of Lp, p > 0, Orlicz, Lorentz, and Köthe spaces. Let ϱ be a pseudomodular, Lϱ the corresponding modular function space, and C a sublattice of Lϱ. Given a function f ∈ Lϱ we consider the minimization problem of finding h ∈ C such that ϱ(f - h) = inf{f - g: g ∈ C}. Such an h is called a best approximant. Problems of finding best approximants are important in approximation theory and probability theory. In the case where C is Lϱ() for some σ-subalgebra of the original σ-algebra, finding best approximants is closely related to the problem of nonlinear prediction. Throughout most of the paper we assume only that ϱ is a pseudomodular and except in one section, we do not assume ϱ to be orthogonally additive. This allows, for instance, application to Lorentz type Lp spaces. If ϱ is a semimodular or a modular, then Lϱ can be equipped with an F-norm ∥ · ∥ϱ and one considers the corresponding F-norm minimization problem. This paper gives several existence theorems relating to this problem, a theorem comparing the set of all best ϱ-approximants with the set of all best ∥ · ∥ϱ-approximants and a uniqueness theorem.

## Department(s)

Mathematics

## Document Type

Article

## DOI

https://doi.org/10.1016/0021-9045(90)90126-b

## Publication Date

1990

## Recommended Citation

Kilmer, Shelby J., Wojciech M. Kozlowski, and Grzegorz Lewicki. "Best approximants in modular function spaces." Journal of Approximation Theory 63, no. 3 (1990): 338-367.

## Journal Title

Journal of Approximation Theory