Continued fractions associated with trigonometric and other strong moment problems

General T-fractions and M-fractions whose approximants form diagonals in two-point Padé tables are subsumed here under the study of Perron-Carathéodory continued fractions (PC-fractions) whose approximants form diagonals in weak two-point Padé 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 Carathéodory 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 Szegö polynomials orthogonal on the unit circle. Szegö 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].

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