ABSTRACT
To a given modified PC-fraction with approximants An/Bn there exists a unique pair (L 0, L ∞) of formal power series at 0 and ∞ such that A 2m /B 2m is the weak (m, m) two-point Padé approximant to L 0 of order m + 1 and to L ∞ of order m, while A 2m+1/B 2m+1 is the weak (m, m) two-point Padé approximant to L 0 of order m and to L ∞ of order m + 1. The canonical denominators of the even contraction of the modified PC-fraction (which is a modified T-fraction) are orthogonal Laurent polynomials obtained from the basis {1, z −1, z, z −2, z 2, … }, while the canonical denominators of the odd contractions of the modified PC-fraction (which is a modified M-fraction) are orthogonal Laurent polynomials obtained from the basis {1, z, z −1, z 2, z −2, … }.
An analogous situation arises when the pair of power series (L 0, L ∞) is replaced by Newton series determined by a general interpolation table of points on the real line, the modified PC-fraction and its contractions are replaced by appropriate analogous continued fractions, and orthogonal Laurent polynomials are replaced by orthogonal rational functions with poles in the set of interpolation points. In this paper an investigation of these relationships is carried out.
