A Characterization of Best φ-Approximants with Applications to Multidimensional Isotonic Approximation

Abstract

Some properties of best monotone approximants in several variables are obtained. We prove the following abstract characterization theorem. Let ( , A, µ) be a measurable space and let L ⊂ A be a σ-lattice. If f belongs to a Musielak–Orlicz space Lϕ( , A, µ), then there exists a σ-algebra Af ⊂ A such that g is a best ϕapproximant to f from Lϕ(L) iff g is a best ϕ-approximant to f from Lϕ(Af ). The σ-algebra Af depends only on f . When ⊂ Rn and Lϕ(L) is the set of monotone functions in several variables, we give sufficient conditions on the geometry of to obtain a uniqueness theorem. This result extends and unifies previous ones. Finally, we prove a coincidence relation between a function and its best ϕ-approximant. Our main results are new, even in the classical Lebesgue spaces Lp