Approximation on partially ordered sets of regular grids

Abstract

In this paper we analyze the approximation of functions on partially ordered sequences of regular grids. We start with the formulation of minimal requirements for useful grid transfer operators. We introduce the notions of nested and of commutative transfer operators. We define mutual coherence for representations on grids that are not related by coarsening or refining. We show necessary and sufficient conditions for mutual coherence and we show how a hierarchical decomposition is generated by a set of commutative transfer operators. The usual piecewise constant and piecewise d-linear approximations are identified as special instances of tensor product type. In the second part of the paper we derive error estimates for approximation in these spaces, in different norms on general d-dimensional dyadic sequences of regular and sparse grids. Some of these results have been published before, e.g., in doctoral theses by Bungartz and Pflaum. Here, the results are presented in a unified framework and the proofs are much simplified. We pay special attention to a convenient notation.