Best Simultaneous Local Approximation in the Lp Norms

Abstract

We study the behavior of the best simultaneous approximation to two functions from a convex set in Lp spaces, 2<p<∞, on a finite union of intervals when its measure tends to zero. In particular, we give sufficient conditions over the differentiability of two functions to assure existence of the best simultaneous local approximation from the class of algebraic polynomials of a fixed degree. These conditions are weaker than the ordinary differentiability given in previous works. More precisely, we consider differentiable functions in the sense Lp.