ON THE CONVERGENCE OF POLYNOMIAL-APPROXIMATION OF RATIONAL FUNCTIONS

This paper investigates the convergence condition :br the polynomial a pproximation of rational functions and rational curves. The main resul t, based on a hybrid expression of rational functions (or curves), is that two-point Hermite interpolation converges if all eigenvalue modul i of a certain r x r matrix are less than 2, where r is the degree of the rational function (or curve), and where the elements of the matrix are expressions involving only the denominator polynomial coefficient s (weights) of the rational function (or curve). As a corollary for th e special case of r = 1, a necessary and sufficient condition for conv ergence is also obtained which only involves the roots of the denomina tor of the rational function and which is shown to be superior to the condition obtained by rite traditional remainder theory for polynomial interpolation. For the low degree cases (r = 1, 2, and 3), concrete c onditions are derived. Application to rational Bernstein-Bezier curves is discussed. (C) 1997 Academic Press.