Rational Carathéodory-Fejér approximation on a disk, a circle, and an interval
A systematic description of the Carathéodory-Fejér method (CF method) is given for near-best uniform rational approximation of a (high degree) polynomial on a disk. On the basis of Takagi's extension of the Carathéodory-Fejér theorem degeneracies are characterized and it is proven that they appear in the CF table, which is introduced, in square blocks. Corresponding methods for complex and real trigonometric rational approximation, and for ordinary real rational approximation on an interval are then derived. For each problem certain weight functions are allowed and several types of CF approximation are defined depending on truncation. In Part II of ''On complex rational approximation'' [in ''Computational Aspects of Complex Analysis'', H. Werner et al., eds., D. Reidel Publ. Co., Dordrecht (1983), pp. 103-132] these results have meanwhile been extended to the case where the given function is a higher degree rational function.
near-best uniform rational approximation
scientific article; zbMATH DE number 3863787
scientific article