Real Polynomial Chebyshev Approximation by the Carathiodory - Fejir Method

A new method is presented for near-best approximation of a real function F on [-r, r] by a polynomial of degree m. The method is derived by transplanting the given problem to the unit disk, then applying the Carath6odory-Fej6r theorem. The resulting near-best approximation is constructed from the principal eigenvalue and eigenvector of a Hankel matrix of Chebyshev coefficients of F. It is well known that as-0, the mth partial sum of the Chebyshev series of F agrees with the best approximation to a relative error O('r). In contrast, our approximation is shown to differ from best by at most O(7"2m+3). m similar result is given for approximation on [-1, 1] as m oe. Such high-order agreement is of both practical and theoretical importance. In particular, it establishes a real analogue of the phenomenon that on the complex unit disk best approximation error curves tend to closely approximate circles. Several numerical examples are presented.