An algebraic approach to multivariate approximation

An algebraic method for the study of certain multivariate approximation techniques is developed and examined. This method, introduced by Gordon [8], employs the properties of a distributive lattice ψ M generated by a set of linear projection operators {P_{m}^Mm=1. ψ has a maximal element M and a minimal element L. The maximal projector M combines all the approximation properties of the entire collection {Pm}. while the minimal projector L has only those properties common to all the Pm= 1, 2,..., M.}

A comparison of the maximal and minimal projectors for several examples is explored. These examples serve to illustrate the applications of this algebraic approach to the study of certain approximation techniques, as well as to provide specific interpolatory meaning to the algebraic properties of ψ. In particular, an example involving the use of cubic splines contrasts the maximal method of successive decomposition with the minimal tensor product technique. Under fairly mild assumptions, the method of successive decomposition is much more efficient (in terms of the number of data points required) than is the tensor product method.

The application of this algebraic approach to other multivariate techniques is easily seen. This approach provides an interesting and useful technique for the study of multivariate approximation.