On the Uniqueness of Best Non-decreasing Approximation in Orlicz Spaces

TL;DR

This paper proves the uniqueness of the best non-decreasing approximation in Orlicz spaces by establishing the continuity of the approximation set for certain convex functions.

Contribution

It introduces a new approach to demonstrate the uniqueness of best monotone approximation in Orlicz spaces using continuity properties.

Findings

Established the continuity of the best monotone approximation set.

Proved the uniqueness of the best non-decreasing approximation in Orlicz spaces.

Applied characterization of approximation sets to convex functions.

Abstract

Given an approximately continuous function $f$ in an Orlicz space $L^\Phi([a,b]),$ for a suitable class of convex functions $\Phi,$ we employ a characterization of the best monotone approximation set to establish its continuity, which in turn yields the uniqueness property for the best monotone approximation in $L^\Phi([a,b]).$