The Natural Best L1-Approximation by Nondecreasing Functions
Journal of Approximation Theory
We construct a candidate for the natural best L1-approximation to an integrable function, ƒ, by elements of an L1-closed convex proximinal set. If ƒ is a Lebesgue integrable function on [0, 1] and the approximating set is the set of all nondecreasing functions, we show that our construction gives an extension of the known natural best L1-approximation operator from ∪p > 1Lp to L1. In the course of doing this, we also complete the characterization, given in (Huotari, Meyerowitz, and Sheard, J. Approx. Theory47(1986), 85–91) of the set of all best L1-approximations. Finally, in the case of isotonic approximation to a function of several variables, we extend a previous result concerning the almost everywhere convergence of the best Lp-approximations, p > 1, to the natural best L1-approximation.
R Huotari, David A. Legg, A Meyerowitz, and Douglas W. Townsend (1988).
The Natural Best L1-Approximation by Nondecreasing Functions. Journal of Approximation Theory.52 (2), 132-140.