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Schur fractions, Perron-Carathéodory fractions and Szegö polynomials, a survey

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Analytic Theory of Continued Fractions II

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1199))

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References

  1. N. Akhiezer, The classical moment problem, Hafner, New York (1965).

    MATH  Google Scholar 

  2. G. Baker and P. Graves-Morris, "Padé Approximants. I, II," Encyclopedia of Mathematics and its Applications, 13,14, Addison-Wesley, Reading, Massachusetts (1980). Now available from Cambridge Univ. Press.

    MATH  Google Scholar 

  3. C. Carathéodory, "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen," Math. Ann. 64 (1907), 95–115.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Carathéodory, "Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen," Rend. Circolo Math. Palermo 32 (1911), 193–217.

    Article  MATH  Google Scholar 

  5. C. Carathéodory and L. Fejer, "Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landau'schen Satz," Rend. Circolo Math. Palermo 32 (1911), 218–239.

    Article  MATH  Google Scholar 

  6. E. Fischer, "Über das Carathéodory'sche Problem, Potenzreihen mit positive reellen Teil betreffend," Rend. Circolo Math. Palermo 32 (1911), 240–256.

    Article  MATH  Google Scholar 

  7. Y. Geronimus, "On the trigonometric moment problem," Ann. of Math. (2) 47 (1946), 742–761.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Hamel, "Eine charakteristische Eigenschaft beschränkter analytischer Funktionen, Math. Ann. 78 (1917), 257–269.

    Article  MathSciNet  MATH  Google Scholar 

  9. W. B. Jones, O. Njåstad and W. J. Thron, "Continued Fractions associated with the trigonometric and other strong moment problems," Constructive Approximation, to appear.

    Google Scholar 

  10. W. B. Jones, O. Njåstad and W. J. Thron, "Szegö polynomials and Perron-Carathéodory fractions," in preparation.

    Google Scholar 

  11. W. B. Jones and A. Steinhardt, "Digital filters and continued fractions," Analytic Theory of Continued Fractions, (W. B. Jones, W. J. Thron and H. Waadeland, Eds.) Lecture Notes in Mathematics, No. 932, Springer, New York (1982), 129–151.

    Chapter  Google Scholar 

  12. W. B. Jones and W. J. Thron, "Continued fractions: Analytic theory and applications," Encyclopedia of Mathematics and its Applications, 11, Addison-Wesley, Reading, Mass. (1980). Now available from Cambridge Univ. Press.

    MATH  Google Scholar 

  13. O. Perron, Lehre von den Kettenbrüchen, Band II, Teubner, Stuttgart (1957).

    MATH  Google Scholar 

  14. J. Schur, "Über Potenzreihen die im Inneren des Einheitskreises beschränkt sind," J. reine angewandte Math. 147 (1917), 205–232, 148 (1918/19), 122–145.

    MathSciNet  MATH  Google Scholar 

  15. J. A. Shohat and J. D. Tamarkin, "The problem of moments," Math. Surveys, No. 1, Amer. Math. Society, Providence, R.I. (1943).

    MATH  Google Scholar 

  16. G. Szegö, Collected Papers, R. Askey Ed., vol. 1, Birhkäuser, Basel, Boston, Stuttgart (1982).

    Google Scholar 

  17. G. Szegö, Orthogonal Polynomials, Amer. Math. Society, Providence, R.I. (1939).

    Book  MATH  Google Scholar 

  18. W. J. Thron, "Two-point Padé tables, T-fractions and sequences of Schur," Padé and Rational Approximation, (E. B. Saff and R. S. Varga, Eds.) Academic Press, New York (1977), 215–226.

    Chapter  Google Scholar 

  19. O. Toeplitz, “Über die Fourier'sche Entwicklung positiver Funktionen," Rend. Circolo Math. Palermo 32 (1911), 191–192.

    Article  MATH  Google Scholar 

  20. H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York (1948).

    MATH  Google Scholar 

  21. H. S. Wall, "Continued fractions and bounded analytic functions," Bull. Amer. Math. Soc. 50 (1944), 110–119.

    Article  MathSciNet  MATH  Google Scholar 

  22. Gregory S. Ammar and William B. Gragg, "The generalized Schur algorithm for the superfast solution of Toeplitz systems," Proceedings of the Conference on Padé Approximation in Lancut, Poland (1985), to appear in the Springer Lecture Notes in Mathematics.

    Google Scholar 

  23. Gregory S. Ammar and William B. Gragg, "The implementation and use of the generalized Schur algorithm," Proceedings of the 7th International Symposium on the Mathematical Theory of Networks and Systems (MTNS-85), Stockholm (C. Byrnes and A. Lindquist, eds.), North Holland, to appear.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Campus Box 426, 80309-0426, Boulder, CO, USA

    W. B. Jones

  2. Department of Mathematics, University of Trondheim, NTH, N-7034, Trondheim, Norway

    O. Njåstad

  3. Department of Mathematics, University of Colorado, Campus Box 426, 80309-0426, Boulder, CO, U.S.A.

    W. J. Thron

Authors
  1. W. B. Jones
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  2. O. Njåstad
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  3. W. J. Thron
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Wolfgang J. Thron

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© 1986 Springer-Verlag

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Jones, W.B., Njåstad, O., Thron, W.J. (1986). Schur fractions, Perron-Carathéodory fractions and Szegö polynomials, a survey. In: Thron, W.J. (eds) Analytic Theory of Continued Fractions II. Lecture Notes in Mathematics, vol 1199. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075938

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  • DOI: https://doi.org/10.1007/BFb0075938

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  • Print ISBN: 978-3-540-16768-6

  • Online ISBN: 978-3-540-38817-3

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