On the convergence of polynomial approximation of rational functions
Based on the notion of hybrid polynomials, the authors derive necessary and sufficient convergence criteria for various polynomial approximations of rational functions and rational curves. This paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-pointed Hermite interpolation converges if all eigenvalue moduli or a certain \(n \times n\) matrix are less than 2, where \(n\) is the degree of the rational function (or curve), and where the elements of the matrix are expressions involving only the denominator polynomial coefficients (weights) of the rational function (or curve). As a corollary for the special case of \(n=1\), a necessary and sufficient condition for convergence is also obtained which only involves the roots of the denominator of the rational function and which is shown to be superior to the condition obtained by the traditional remainder theory for polynomial interpolation. For the low degree case \((n= 1,2\), and 3), concrete conditions are derived. Application to rational Bernstein-Bézier curves is also discussed.
scientific article; zbMATH DE number 1033752
scientific article