ON POLYNOMIAL IDEALS OF FINITE CODIMENSION WITH APPLICATIONS TO BOX SPLINE THEORY

We investigate the relation between an ideal I of finite codimension in the space π of multivariate polynomials, and ideals which are generated by lower order perturbations of some generators for I. Of particular interest are the codimension of these ideals and the local approximation order of their kernels. The discussion, stimulated by recent results in approximation theory, allows us to provide a simple analysis of the polynomial and exponential spaces associated with box splines. We describe their structure, dimension, and local approximation order and an algorithm for their construction. The resulting theory is extended to subspaces of the above exponential/polynomial spaces. © 1991.