Norms of projectors onto spaces with Riesz bases

Abstract

Many important linear operators P:X→ S of a linear space X onto a subspace S of X are defined by the minimum problem ∥f− Pf∥= min{∥f−u∥ : u∈ S} , f∈ X, where the norm ∥·∥ on X is induced by an inner product. We study the L ∞-norm of such operators P when S has a Riesz basis B . Then we apply the results in detail to the best L 2- and ℓ 2-approximations for univariate polynomial splines spaces and to Box spline series.