Abstract
Perron-Carathéodory continued fractions (PC-fractions) have recently been investigated in connection with the trigonometric moment problem and Szegö polynomials (orthogonal on the unit circle) [5] and with Wiener's linear prediction method used in digital signal processing [4]. Further properties of PC-fractions are developed here. These include: fast algorithms for computing PC-fractions, connections with other strong moment problems (Stieltjes and Hamburger) and the relationship to the more general class of Perron continued fractions.
Research supported in part by the U.S. National Science Foundation under Grant No. DMS-8401717.
Research supported in part by grants from the United States Educational Foundation in Norway (Fulbright-Hays Grant), The Norwegian Marshall Fund and the University of Colorado Council on Research and Creative Work.
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References
Bultheel, Adhemar, Algorithms to compute the reflection coefficients of digital filters, Numerical Methods of Approximation Theory, Vol. 7, (eds. L. Collatz, G. Meinardus, H. Werner) Birkhäuser Verlag, Basel (1984), 33–50.
McCabe, J.H., On the even extension of an M-fraction, in: Padé Approximation and Its Applications (eds. M.G. deBruin and H. van Rossum), Lecture Notes in Mathematics 888, Springer-Verlag, New York (1980), 290–299.
Jones, William B., Olav Njåstad and W.J. Thron, Orthogonal Laurent polynomials and the strong Hamburger moment problem, J. Math. Anal. and Applic., 98, No. 2 (February 1984), 528–554.
Jones, William B., Olav Njåstad and W.J. Thron, Continued fractions associated with Wiener's linear prediction method, Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, Stockholm, Sweden in June 1985, North Holland, to appear.
Jones, William B., Olav Njåstad and W.J. Thron, Continued fractions associated with the trigonometric and other strong moment problems, submitted.
Jones, William B. and Thron, W.J., Continued fractions: Analytic theory and applications, Encyclopedia of Mathematics and Its Applications, No. 11, Addison-Wesley Publishing Company, Reading, Mass. (1980).
Jones, William B., W.J. Thron, and H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc., V. 261, No. 2 (October, 1980), 503–528.
Levinson, Norman, The Wiener RMS (root mean square) error criterion in filter design and prediction, J. Math. and Physics 25 (1947), 261–278.
Magnus, Arne, On the structure of the two-point Padé table, Analytic theory of continued fractions, (eds. W.B. Jones, W.J. Thron, H. Waadeland), Lecture Notes in Mathematics 932, Springer-Verlag, New York (1982), 176–193.
Perron, O., Die Lehre von den Kettenbrüchen, Band II, B.G. Teubner, Stuttgart (1957).
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Jones, W.B., Njåstad, O., Thron, W.J. (1987). Perron-Carathéodory continued fractions. In: Gilewicz, J., Pindor, M., Siemaszko, W. (eds) Rational Approximation and its Applications in Mathematics and Physics. Lecture Notes in Mathematics, vol 1237. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072464
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DOI: https://doi.org/10.1007/BFb0072464
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