A Family of Natural Best L1-Approximants by Nondecreasing Functions
Numerical Functional Analysis and Optimization
1 & 2
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 ∞.
David A. Legg and Douglas W. Townsend (1992).
A Family of Natural Best L1-Approximants by Nondecreasing Functions. Numerical Functional Analysis and Optimization.13 (1 & 2), 173-183.