Branch Points, M Fractions and Rational Approximants Generated by Linear Equations

Drew, DM; Murphy, JA

doi:10.1093/imamat/19.2.169

Cited by 6 publications

(1 citation statement)

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“…where H~ ") := k-1 det(Cm+~,+~),,.~=o, m=0,+l,+2,.., and k=1,2,3,.., are the Hankel determinants associated with {c,}_~oo, H(o"): = 1. general T-fraction representations of important special functions can be found in [6], [7], [8], [9], [23], [27], [28], [31], [35], [40] and [41]. Applications to Volterra integral equations and Laplace transforms are considered in [14], [15], and [16].…”

Section: Introductionmentioning

confidence: 99%

Continued fractions associated with trigonometric and other strong moment problems

1986

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General T-fractions and M-fractions whose approximants form diagonals in two-point Pad6 tables are subsumed here under the study of Perron-Carath6odory continued fractions (PC-fractions) whose approximants form diagonals in weak two-point Pad6 tables. The correspondence of PCfractions 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 Carath6odory 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 Szeg6 polynomials orthogonal on the unit circle. Szeg6 polynomials are shown to be the odd ordered denominators of positive PC-fractions. Positive PCfractions are also related to Wiener filters used in digital signal processing [3], [251.

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