Real Polynomial Chebyshev Approximation by the Carathéodory–Fejér 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 thephenomenon that on the complex unit disk best approximation error curves tend to closely approximate circles. Several numerical examples are presented. Introduction. Two main ideas are combined in this paper. The first is that by means of the Joukowski map x =1/2(w + w-a), the real Chebyshev approximation problem on the unit interval (-1, 1 can be related to a complex Chebyshev approxima- tion problem on the unit disk Iwl< 1. Under this transplantation near-best approxima- tions for one problem often correspond to near-best approximations for the other, and so an approximation method for one problem can be carried over to the other. The second is that exceedingly good near-best approximations on the unit disk can be computed by an application of the Carath6odory-Fej6r theorem, which involves the principal singular value and singular vector of a Hankel matrix of Taylor series coefficients. Transplanting this technique to (-1, 1) leads to a powerful real near-best approximation method that is based upon the principal eigenvalue and eigenvector of a Hankel matrix of Chebyshev series coefficients of the function to be approximated. Both of these ideas take advantage of the phenomenon that in approximation on the unit disk best approximation error curves tend to closely approximate perfect circles. The complex "CF method" was developed by Trefethen in response to this phenomenon, partly to explain and partly to exploit it (18), (19). The transplantation technique was proposed in connection with Pad6 approximation by Frankel, Gragg and Johnson (5), (7), and has been extended by Gutknecht (8). Combining the two leads to the conclusion that a real best approximation error curve not only equioscil- lates, but also approximates the real part of a multiply winding Blaschke product in the complex plane. In fact, the great majority of analytically known examples of Chebyshev approximations are either complex examples with perfectly circular error curves (finite Blaschke products), or transplantations of these to a real interval. Such examples will be discussed in a unified way by Gutknecht in a forthcoming paper. Both the Joukowski transplantation and the CF theory extend to rational approxi- mation; for the latter see (9) and (19). However, as the rational theory is more complicated and as some of its results are more limited, this paper is confined to the polynomial case. Section 1 presents the CF method. Section 2 establishes two a