A family of natural best L₁-approximants by nondecreasing functions

Abstract

Let fbe a bounded function on [0, 1] and f _p be the best L _p approximant to fby nondecreasing functions. The usual natural bestLi-approximant fi is characterized by f ₁ - lim _p→1 f _p . In this paper we consider families of norms induced by Orlicz functions ϕ _a (x) such that ϕ _a (x) → x as a → 0 but for which the best L _{ϕ a} -approximants f _{ϕ a} to fconverge to functions other than the natural best L ₁-approximant. We give an example to show that in some cases, any best L ₁ approximant to f can be thought of as the natural best with respect to some family of norms. Finally, an unbounded f and a family of norms induced by ϕ _a (x) are given such that f _{ϕ a} diverges in L ₁norm to ∞.