Efficient evaluation of 1-D and 2-D polynomials at equispaced points

An extension to the two-dimensional case of a previously published fast algorithm for evaluating exponential functions and polynomials over periodically distributed points is generalized to evaluate two-dimensional polynomials efficiently over general periodic parallelograms. The problem of getting the initial values required for other fast algorithms is treated in a general manner. Fast algorithms are presented for computing initial values in both cases of one- and two-dimensional polynomials. It is observed that the two-dimensional recursion is not always advantageous compared with taking the one-dimensional recursion for each variable separately