Chebyshev Series Expansion of Inverse Polynomials

Abstract

An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f(x)/sum_(j=0..k)b_j*T_j(x)=1+sum_(n=k+1..oo) a_n*T_n(x), and may be handled with a Newton method providing the Chebyshev expansion of f(x) is known.