Approximation of continuous and quasi-continuous functions by monotone functions

Abstract

Let Q denote the Banach space (sup norm) of quasi-continuous functions defined on the interval [0, 1]. Let C denote the subspace comprised of continuous functions. Met M denote the closed convex cone in Q comprised of nondecreasing functions. For f ϵ Q and 1 < p < ∞, let f p denote the best L p -approximation to f by elements of M. It is shown that f p converges uniformly as p → ∞ to a best L ∞-approximation to f by elements of M. If f ϵ C, then each f p ϵ C; so f ∞ ϵ C.