Interpolation and best simultaneous approximation

Abstract

We consider best simultaneous approximation to k continuous functions on an interval [a,b] from a finite dimensional subspace of C[a,b], with respect to the functionals σj=1kψ(∫abΦ(|fj|)) and max1≤j≤k∫abΦ(|fj|) for suitable real functions Φ; and ψ. We obtain the interpolation properties of the best simultaneous approximations. As a consequence, we extend known results in Lp-approximation over small intervals.