Chebyshev series expansion of inverse polynomials

摘要

The Chebyshev series expansion ∑′n=0∞anTn(x) of the inverse of a polynomial ∑j=0kbjTj(x) is well defined if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the an are linear combinations of simple functions of the polynomial roots. Also, if the first k of the coefficients an are known, the others become linear combinations of these derived recursively from the bj's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the bj in f(x)/∑0kbjTj(x)=1+∑k+1∞anTn(x); a Newton algorithm produces these if the Chebyshev expansion of f(x) is known.