Best Simultaneous approximation of quasi-continuous functions by monotone functions

Abstract

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.