Approximation on partially ordered sets of regular grids

In this paper we analyse the approximation of functions on partially ordered sequences of regular grids. We start with the formulation of minimal requirements for useful grid transfer operators in such a partially ordered context, and we continue with the introduction of hierarchical decompositions and the identification of piecewise constant and piecewise linear approximations as special instances of the tensor product case. In the second part of the paper we derive error estimates for approximation in different norms on more-dimensional dyadic sequences of regular and sparse grids. We give special attention to a convenient notation.