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 L p , p > 0, Orlicz, Lorentz, and Köthe spaces. Let ϱ be a pseudomodular, L ϱ the corresponding modular function space, and C a sublattice of L ϱ . Given a function f ∈ L ϱ 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 L ϱ( B ) for some σ-subalgebra B 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 L p spaces. If ϱ is a semimodular or a modular, then L ϱ 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.