Interpolation and best simultaneous approximation

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