L _ø-Approximation by nondecreasing functions on the interval

Abstract

Letφ: [0, ∞) → [0, ∞) be a Δ₂ convex function,φ(0)=0,φ (y)>0 ify>0, and letf be a Lebesgue measurable function defined on [0, 1], ∫ ¹₀ φ(|f|)<∞. We consider the setM _ø (f) of elements that minimize ∫ ¹₀ φ(|f−g|) among the class of nondecreasing functionsg. We give a description of that set, showing that there exists an open setU such that any functiong ₀ εM _ø(f) is constant on each component ofU. Furthermore, whenever the right derivativeφ’₊ is bounded, orf is essentially bounded, then anyg ₀ coincides withf almost everywhere outsideU, and we show with an example that the former hypotheses are essential for getting that result. We also give a construction of the minimum and maximum elements inM _ø(f). Finally, we prove that iff is approximately continuous at every point, then there is a uniqueg ₀ εM _ø, andg ₀ is continuous. It is also observed that analogous results are valid when approximating with respect to the Luxemburg norm in the associated Orlicz space.