Norms of projectors onto spaces with Riesz bases

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 L2- and ℓ2-approximations for univariate polynomial splines spaces and to Box spline series.