Continued fractions associated with trigonometric and other strong moment problems

Abstract

General T-fractions and M-fractions whose approximants form diagonals in two-point Pade tables are subsumed here under the study of Perron-Caratheodory continued fractions (PC-fractions) whose approximants form diagonals in weak two-point Pade tables. The correspondence of PC-fractions with pairs of formal power series is characterized in terms of Toeplitz determinants. For the subclass of positive PC-fractions, it is shown that even ordered approximants converge to Caratheodory functions. This result is used to establish sufficient conditions for the existence of a solution to the trigonometric moment problem and to provide a new starting point for the study of Szego polynomials orthogonal on the unit circle. Szego polynomials are shown to be the odd ordered denominators of positive PC-fractions. Positive PC-fractions are also related to Wiener filters used in digital signal processing [3], [25].