Spline spaces and Toeplitz operators

Abstract

This chapter presents the basic theory of Toeplitz operators on l ^p-spaces as well as of spline approximation methods for singular integral operators with constant coefficients, and it highlights the fruitful interplay between these two fields. The first fundamental result is de Boor’s estimate which shows that spline spaces can be viewed as embeddings of thel^p -spaces into L^p-spaces. This allows to think of Toeplitz operators previously acting on l^p -spaces as acting on subspaces of L^p, and it offers the applicability of (continuous) Fourier and Mellin techniques to the (discrete) Toeplitz operators which will be done in the first part of the chapter. In its second part we shall employ the theory of Toeplitz operators to derive results on spline approximation methods. For instance, we shall explain how the local behaviour of Toeplitz operators determines such things as the convergence of certain approximation sequences in the strong operator topology. The basic observations in this chapter are four theorems (Lemma 2.1, Theorem 2.5, Proposition 2.17, Theorem 2.7) whose proofs are unfortunately rather technical and not very instructive. For that reason we have separated these proofs from the other material and present them in a separate section which can be omitted in the first reading without loss.