On Polynomial Ideals of Finite Codimension with Applications to Box Spline Theory

We investigate here the relations between an ideal I of finite codimension in the space of multivariate polynomials and various ideals which are generated by lower order perturbations of the generators of I. Special emphasis is given to the question of the codimension of I and its perturbed counterpart and to the local approximation order of their kernels. The discussion, stimulated by certain results in approximation theory, allows us to provide a simple analysis of the polynomial and exponential spaces associated with box splines. This includes their structure, dimension, local approximation order and an algorithm for their construction. The resulting theory is extended to subspaces of the above exponential/polynomial spaces.

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