Best Approximants in Musielak-Orlicz Spaces


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(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 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.



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Journal of Approximation Theory