Skip to main content
SpringerLink
Log in

Continued fractions associated with trigonometric and other strong moment problems

  • Published:
Constructive Approximation Aims and scope

Abstract

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].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. I. Akheizer (1965): The Classical Moment Problem, (translated by N. Kummer). New York: Hafner.

    Google Scholar 

  2. G. A. Baker, Jr., G. S. Rushbrooke, H. E. Gilbert (1964):High temperature series expansions for the spin −1/2 Heisenberg model by the method of irreducible representations of the symmetry group. Phys. Rev.,135 (5A):A1272-A1277.

    Google Scholar 

  3. A. Bultheel (1984):Algorithms to compute the reflection coefficients of digital filters. In: Numerical Methods of Approximation Theory, Vol. 7 (L. Collatz, G. Meinardus, H. Werner, eds.), Basel: Birkhäuser-Verlag, pp. 33–50.

    Google Scholar 

  4. J. Čizek (1982):Large order perturbation theory in the context of atomic and molecular physics-interdisciplinary aspects. Internat. J. Quantum Chem.,XXI:27–68.

    Google Scholar 

  5. J. Čizek, A. Pellegatti J. Paldus (1975):Correlation effects in the PPP Model of alternate π-electronic systems: Two-point Padé approximant approach. Internat. J. Quantum Chem.,IX:987–1007.

    Google Scholar 

  6. D. M. Drew, J. A. Murphy (1977):Branch points, M-fractions and rational approximants generated by linear equations. J. Inst. Math. Appl.,19:169–185.

    Google Scholar 

  7. W. Gautschi (1967):Computational aspects of three-term recurrence relations. SIAM Rev.,9 (1):24–82.

    Google Scholar 

  8. W. Gautschi (1977):Anomalous convergence of a continued faction for ratios of Kummer functions. Math. Comp.,31 (140):994–999.

    Google Scholar 

  9. W. Gautschi, J. Slavik (1978):On the computation of modified Bessel function ratios. Math. Comp.,32 (143):865–875.

    Google Scholar 

  10. J. Geronimus (1946):On the trigonometric moment problem, Ann. of Math.47, No. 4, 742–761.

    Google Scholar 

  11. Ya. L. Geronimus (1954):Polynmials orthogonal on a circle and their applications. Amer. Math. Soc. Trans., No. 104.

  12. W. B. Gragg (1980):Truncation error bounds for T-fractions. In: Approximation Theory III, New York: Academic Press, pp. 455–460.

    Google Scholar 

  13. U. Grenander, G. Szegö (1958): Toeplitz Forms and Their Applications. Berkeley: University of California Press.

    Google Scholar 

  14. R. E. Grundy (1977):Laplace transform inversion using two-point rational approximants. J. Inst. Math. Appl.,20:299–306.

    Google Scholar 

  15. R. E. Grundy (1978a):The solution of Volterra integral equations of the convolution type using two-point rational approximants. J. Inst. Math. Appl.,22:147–158.

    Google Scholar 

  16. R. E. Grundy (1979b):On the solution of non-linear Volterra integral equations using two-point Padé rational approximants. J. Inst. Math. Appl.,22:317–320.

    Google Scholar 

  17. H. Hamburger (1920/21):Über eine Erweiterung des Stieltjesschen Momenten problems, Parts I, II, III. Math. Ann.,81:235–319;82:120–164, 168–187.

    Google Scholar 

  18. G. Hamel (1918):Eine charakteristische Eigenschaft beschränkter analytischer Funktionen. Math. Ann.,78:257–269.

    Google Scholar 

  19. P. Henrici (1974): Applied and Computational Complex Analysis, vol. 1. Power Series, Integration, Conformal Mapping and Locations of Zeros. New York: Wiley.

    Google Scholar 

  20. K. Hoffman, R. Kunze (1971): Linear Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  21. A. Isihara, E. W. Montroll (1971):A note on the ground state energy of an assembly of interacting electrons. Proc. Nat. Acad. Sci. U.S.A.,68 (12):3111–3115.

    Google Scholar 

  22. W. B. Jones (1977):Multiple point Padé tables. In: Padé and Rational Approximation (E. B. Saff and R. S. Varga, eds.), New York: Academic Press, pp. 163–171.

    Google Scholar 

  23. W. B. Jones, O. Njåstad, W. J. Thron (1983a):Two-point Padé expansions of a family of analytic functions. J. Comput. Appl. Math.,9:105–123.

    Google Scholar 

  24. W. B. Jones, O. Njåstad, W. J. Thron (1983b):Continued fractions and the strong Hamburger moment problem. Proc. London Math. Soc., (3),47:363–384.

    Google Scholar 

  25. W. B. Jones, A. Steinhardt (1984):Applications of Schur fractions to digital filtering and signal processing. In: Rational Approximation and Interpolation (P. Graves-Morris, E. B. Saff, R. S. Varga, eds.), Lecture Notes in Mathematics, 1105. Berlin: Springer-Verlag, pp. 210–226.

    Google Scholar 

  26. W. B. Jones, W. J. Thron (1977):Two-point Padé tables and T-fractions. Bull. Amer. Math. Soc.,83:388–390.

    Google Scholar 

  27. W. B. Jones, W. J. Thron (1980): Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and Its Applications, 11. Reading, MA: Addison-Wesley, distributed now by Cambridge University Press, New York.

    Google Scholar 

  28. W. B. Jones, W. J. Thron (1985):On the computation of incomplete gamma functions in the complex domain. J. Comput. Appl. Math.,12 & 13:401–417.

    Google Scholar 

  29. W. B. Jones, W. J. Thron, H. Waadeland (1980):A strong Stieltjes moment problem. Trans. Amer. Math. Soc.,261, No. 2:503–528.

    Google Scholar 

  30. A. Magnus (1982):On the structure of the two-point Padé table. In; Analytic Theory of Continued Fractions (W. B. Jones, W. J. Thron, H. Waadeland, eds.). Lecture Notes in Mathematics, 932. New York: Springer-Verlag.

    Google Scholar 

  31. J. H. McCabe (1974):A continued fraction expansion with a truncation error estimate for Dawson's integral. Math. Comp.,28 (127):811–816.

    Google Scholar 

  32. J. H. McCabe (1975):A formal extension of the Padé table to include two point Padé quotients. J. Inst. Math. Appl.,15:363–372.

    Google Scholar 

  33. J. H. McCabe (1980):On the even extension of an M-fraction. In: Padé Approximation and Its Applications (M. G. de Bruin, H. Van Rossum, eds.). Lecture Notes in Mathematics, 888. New York: Springer-Verlag, pp. 290–299.

    Google Scholar 

  34. J. H. McCabe, J. A. Murphy (1976):Continued fractions which correspond to power series expansions at two points. J. Inst. Math. Appl.,17:233–247.

    Google Scholar 

  35. O. Perron (1957): Die Lehre von den Kettenbrüchen, Band II. Stuttgart: Teubner.

    Google Scholar 

  36. I. Schur (1917/18):Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math.,147 and 148.

  37. P. Sheng (1974):Application of two-point Padé approximants to some solid state problems. Rocky Mountain J. Math.,4 (2):385–386.

    Google Scholar 

  38. P. Sheng, J. D. Dow (1971):Intermediate coupling theory: Padé approximants for polarons. Phys. Rev.,B4:1343–1359.

    Google Scholar 

  39. T. J. Stieltjes (1894):Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Math.,8:J, 1–122;9:A, 1–47; Ouvres,2:402–566.

    Google Scholar 

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

    Google Scholar 

  41. P. Wynn (1959):Converging factors for continured fractions. Numer. Math.,A:272–320.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    William B. Jones & W. J. Thron

  2. University of Trondheim-NTH, 7034, Trondheim, Norway

    Olav Njåstad

Authors
  1. William B. Jones
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olav Njåstad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. J. Thron
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by William B. Gragg.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jones, W.B., Njåstad, O. & Thron, W.J. Continued fractions associated with trigonometric and other strong moment problems. Constr. Approx 2, 197–211 (1986). https://doi.org/10.1007/BF01893426

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01893426

AMS classification

Key words and phrases

Search

Navigation