Best Simultaneous $L^p$ Approximation in the "Sum" Norm

Abstract

In this paper we consider best simultaneous approximation byalgebraic polynomials respect to the norm $\sum_{j=1}^k\|f_j-P\|_p$, $1\le p<\infty$. We prove an interpolation propertyof the best simultaneous approximations and we study the structureof the set of cluster points of the best simultaneousapproximations on the interval $[-\epsilon,\epsilon],$ as $\epsilon \to 0$.