Skip to main content

A survey of bounds for the zeros of analytic functions obtained by continued fraction methods

  • Survey Papers
  • Conference paper
  • First Online:
Rational Approximation and its Applications in Mathematics and Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1237))

  • 431 Accesses

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.G. de Bruin, E.B. Saff, R.S. Varga, On the zeros of generalized Bessel polynomials. I,II, Nederl.Akad.Wetensch. Indag.Math. 43 (1981), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  2. M.G. de Bruin, Generalized Padé tables and some algorithms therein. In: Proceedings of the 1st French-Polish meeting on Padé approximation and convergence acceleration techniques. CPT-81/PE. 1354, Centre de Physique Théorique Marseille, 1982

    Google Scholar 

  3. M.G. de Bruin, Zeros of polynomials generated by 4-term recurrence relations. In: Proceedings of a Conference on Rational Approximation and Interpolation, Springer-Verlag, Berlin-New York, 1984, 331–345

    Chapter  Google Scholar 

  4. ", Simultaneous Padé approximation and orthogonality. In: Polynômes Orthogonaux et Applications, Proceedings, Bar-le-Duc 1984, Lecture Notes in Mathematics No. 1171, Springer-Verlag, Berlin-New York, 1985, 74–83

    Chapter  Google Scholar 

  5. ", On the zeros of recursively defined polynomials, Analysis 6 (1986), 227–236

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Dočev, On the generalized Bessel polynomials, Bulgar.Akad.Nauk. Izv. Mat.Inst. 6 (1962), 89–94 (in Bulgarian)

    MathSciNet  Google Scholar 

  7. E.A. Van Doorn, A note on orthogonal polynomials and oscillation criteria for second order linear difference equations, Memorandum Nr. 522, Technische Hogeschool Twente, May 1985

    Google Scholar 

  8. ", Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tridiagonal matrices, Memorandum Nr. 525, Technische Hogeschool Twente, June 1985.

    Google Scholar 

  9. G. Frobenius, Über Relationen zwischen den Näherungsbrüchen von Potenzreihen, J. für Mathematik, 90 (1881), 1–17

    MathSciNet  Google Scholar 

  10. J. Gilewicz, Approximants de Padé, Lecture Notes in Math.No.667, Springer-Verlag, Berlin, New-York, 1978

    MATH  Google Scholar 

  11. J. Gilewicz, E. Leopold, Location of the zeros of polynomials satisfying three-term recurrence relations. I. General case with complex coefficients, J. Approx. Theory 43 (1985), 1–14

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Gilewicz, E. Leopold, Location of the zeros of polynomials satisfying three-term recurrence relations. II. General case with complex coefficients, in praparation

    Google Scholar 

  13. J. Gilewicz, Sur l'amélioration des théorèmes de localisation des zéros de polynômes, Congrès d'Analyse Numérique 1984, Bombannes (France)

    Google Scholar 

  14. J. Gilewicz, E. Leopold, Fine optimization of the zero-free region for the polynomials satisfying three-term recurrence relations, submitted to J. Approx. Theory

    Google Scholar 

  15. J. Gilewicz, E. Leopold, On the sharpness of results in the theory of location of zeros of polynomials defined three-term recurrence relations. In: Polynômes Orthogonaux et Applications, Proceedings, Bar-le-Duc 1984, Lecture Notes in Math. No. 1171, Springer, Berlin, 1985, 259–266.

    Google Scholar 

  16. E. Grosswald, Besselpolynomials, Lecture Notes in Math. No. 698, Springer-Verlag, Berlin, New-York, 1978

    Google Scholar 

  17. P. Henrici, Note on a theorem of Saff and Varga, Padé and Rational Approximation, Theory and Applications, Academic Press, New York, 1977, 157–161.

    Google Scholar 

  18. A.S. Householder, The theory of matrices in numerical analysis, Dover publications, Inc., New York, 1975

    MATH  Google Scholar 

  19. W.B. Jones, W.J. Thron, Analytic Theory and Applications, Encycl. of Math. and its Applics. Vol. 11, London-Amsterdam — Don Mills, Ontario-Sydney-Tokyo, Addison-Wesley Publ. Co. 1980

    MATH  Google Scholar 

  20. E. Leopold, Approximants de Padé pour les fonctions de classes S, et localisation des zéros de certains polynômes, Thesis, Université d'Aix-Marseille I, 1982

    Google Scholar 

  21. E. Leopold, Location of the zeros of polynomials satisfying three-term recurrence relations. III Positive coefficients case, J. Approx. Theory 43 (1985), 15–24

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Leopold, Location of the zeros of polynomials satisfying three-term recurrence relations. IV Application to some polynomials and to generalized Bessel polynomials, in preparation

    Google Scholar 

  23. D.J. Newman, T.J. Rivlin, The zeros of partial sums of the exponential function, J. Approx. Theory, 5 (1972), 405–412.

    Article  MathSciNet  MATH  Google Scholar 

  24. ", Correction: The zeros of the partial sums of the exponential function, J. Approx. Theory 16 (1976), 299–300.

    Article  MathSciNet  MATH  Google Scholar 

  25. O. Perron, Die Lehre von den Kettenbrüchen, 3 rd ed., vol.2, Teubner, Stuttgart, 1957

    MATH  Google Scholar 

  26. H.-J. Runckel, Zero-free parabolic regions for polynomials with complex coefficients, Proc. AMS 88 (1983), 299–304

    Article  MathSciNet  MATH  Google Scholar 

  27. ", Zero-free regions for polynomials with applications to Padé-approximants, In: Constructive Theory of Functions, Proceedings of the International Conference on constructive theory of functions, Varna 1984, Publishing house of the Bulgarian Academy of Sciences, Sofia 1984, 767–771.

    Google Scholar 

  28. ", Zeros of complex orthogonal polynomials, In: Polynômes, Orthogonaux et Applications, Proceedings, Bar-le-Duc 1984, Lecture Notes in Mathematics, No. 1171, Springer-Verlag, Berlin, New York, 1985, 278–282.

    Google Scholar 

  29. ", Pole-and zero-free regions for analytic continued fractions, Proc. AMS, (1986), Proc. AMS 97 (1986), 114–120.

    Article  MathSciNet  MATH  Google Scholar 

  30. ", Bounds for the eigenvalues of complex tridiagonal matrices, Analysis 6 (1986), 251–253.

    Article  MathSciNet  MATH  Google Scholar 

  31. E.B. Saff, R.S. Varga, On the zeros and poles of Padé approximants to ez, Numer. Math. 25 (1975), 1–14

    Article  MathSciNet  MATH  Google Scholar 

  32. ", Zero-free parabolic regions for sequences of polynomials, SIAM J. Math. Anal. 7 (1976), 344–357

    Article  MathSciNet  MATH  Google Scholar 

  33. ", On the sharpness of theorems concerning zero-free regions for certain sequences of polynomials, Numer. Math. 26 (1976), 345–354

    Article  MathSciNet  MATH  Google Scholar 

  34. H.S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Jacek Gilewicz Maciej Pindor Wojciech Siemaszko

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

De Bruin, M.G., Gilewicz, J., Runckel, HJ. (1987). A survey of bounds for the zeros of analytic functions obtained by continued fraction methods. 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/BFb0072450

Download citation

  • DOI: https://doi.org/10.1007/BFb0072450

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17212-3

  • Online ISBN: 978-3-540-47412-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics