Skip to main content
SpringerLink
Log in

Some Asymptotics for Extremal Polynomials

  • Conference paper
  • First Online:
Book cover Computational Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 155))

  • 595 Accesses

Abstract

We review some asymptotics for Chebyshev polynomials and orthogonal polynomials. Our main interest is in the behaviour of Widom factors for the Chebyshev and the Hilbert norms on small sets such as generalized Julia sets.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. N. Achyeser, Über einige Funktionen, welche in zwei gegebenen Interwallen am wenigsten von Null abweichen. I Teil, Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na (9), 1163–1202 (1932)

    Google Scholar 

  2. N. Achyeser, Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen. II Teil, Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na (3), 309–344 (1933)

    Google Scholar 

  3. G. Alpan, Chebyshev polynomials on generalized Julia sets. Comput. Methods Funct. Theory (2015). doi:10.1007/s40315-015-0145-8

    Google Scholar 

  4. G. Alpan, A. Goncharov, Two measures on Cantor sets. J. Approx. Theory 186, 28–32 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Alpan, A. Goncharov, Orthogonal polynomials for the weakly equilibrium Cantor sets. Proc. Am. Math. Soc. (accepted for publication)

    Google Scholar 

  6. G. Alpan, A. Goncharov, Orthogonal polynomials on generalized Julia sets (preprint). arXiv: 1503.07098v3

    Google Scholar 

  7. G. Alpan, A. Goncharov, Widom Factors for the Hilbert Norm, vol. 107 (Banach Center Publications), pp. 9–16

    Google Scholar 

  8. V.V. Andrievskii, Chebyshev Polynomials on a system of continua. Constr. Approx. (2015). doi:10.1007/s00365-015-9280-8

    MATH  Google Scholar 

  9. A.I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices. Mat. Sb. 125, 231–258 (1984). English translations in Math. USSR Sb. 53 (1986), 233–260

    Google Scholar 

  10. M.F. Barnsley, J.S. Geronimo, A.N. Harrington, Orthogonal polynomials associated with invariant measures on Julia sets. Bull. Am. Math. Soc. 7, 381–384 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. M.F. Barnsley, J.S. Geronimo, A.N. Harrington, Infinite-dimensional Jacobi matrices associated with Julia sets. Proc. Am. Math. Soc. 88, 625–630 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. M.F. Barnsley, J.S. Geronimo, A.N. Harrington, Almost periodic Jacobi matrices associated with Julia sets for polynomials. Commun. Math. Phys. 99, 303–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Brück, M. Büger, Generalized iteration. Comput. Methods Funct. Theory 3, 201–252 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. J.S. Christiansen, Szegő’s theorem on Parreau-Widom sets. Adv. Math. 229, 1180–1204 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.S. Christiansen, B. Simon, M. Zinchenko, Finite gap Jacobi matrices. I. The isospectral torus. Constr. Approx. 32, 1–65 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. J.S. Christiansen, B. Simon, M. Zinchenko, Finite gap Jacobi matrices, II. The Szegö class. Constr. Approx. 33, 365–403 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.S. Christiansen, B. Simon, M. Zinchenko, Finite gap Jacobi matrices: a review, in Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, RI, 2013), pp. 87–103

    MATH  Google Scholar 

  18. J.S. Christiansen, B. Simon, M. Zinchenko, Asymptotics of Chebyshev polynomials, I. Subsets of R (2015). http://arxiv.org/pdf/1505.02604v1.pdf

  19. J.S. Geronimo, W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping. Trans. Am. Math. Soc. 308, 559–581 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Goncharov, Weakly equilibrium Cantor type sets. Potential Anal. 40, 143–161 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Goncharov, B. Hatinoğlu, Widom factors. Potential Anal. 42, 671–680 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. S.M. Heilman, P. Owrutsky, R. Strichartz, Orthogonal polynomials with respect to self-similar measures. Exp. Math. 20, 238–259 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Krüger, B. Simon, Cantor polynomials and some related classes of OPRL. J. Approx. Theory 191, 71–93 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. N.G. Makarov, A.L. Volberg, On the harmonic measure of discontinuous fractals. LOMI Preprint, E-6-86, Steklov Mathematical Institute, Leningrad (1986)

    Google Scholar 

  25. G. Mantica, A stable Stieltjes technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measures. Constr. Approx. 12, 509–530 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Martínez, The spectrum of periodic Jacobi matrices with slowly oscillating diagonal terms. Proc. Edinb. Math. Soc. 51, 751–763 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Milnor, Dynamics in One Complex Variables. Annals of Mathematics Studies, vol. 160 (Princeton University Press, Princeton, NJ, 2006)

    Google Scholar 

  28. F. Peherstorfer, Orthogonal and extremal polynomials on several intervals. J. Comput. Appl. Math. 48, 187–205 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. F. Peherstorfer, P. Yuditskii, Asymptotic behavior of polynomials orthonormal on a homogeneous set. J. Anal. Math. 89, 113–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Ransford, Potential Theory in the Complex Plane (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  31. K. Schiefermayr, A lower bound for the minimum deviation of the Chebyshev polynomial on a compact real set. East J. Approx. 14, 223–233 (2008)

    MathSciNet  MATH  Google Scholar 

  32. B. Simon, Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1, 713–772 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Simon, Szegő’s Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials (Princeton University Press, Princeton, NJ, 2011)

    MATH  Google Scholar 

  34. H. Stahl, V. Totik, General Orthogonal Polynomials. Encyclopedia of Mathematics, vol. 43 (Cambridge University Press, New York, 1992)

    Google Scholar 

  35. G. Szegő, Orthogonal Polynomials, 4th edn. (American Mathematical Society, Providence, RI, 1975). American Mathematical Society, Colloquium Publications, vol. XXIII

    Google Scholar 

  36. J.P. Thiran, C. Detaille, Chebyshev polynomials on circular arcs and in the complex plane, in Progress in Approximation Theory (Academic, Boston, MA, 1991), pp. 771–786

    Google Scholar 

  37. V. Totik, Orthogonal polynomials. Surv. Approx. Theory 1, 70–125 (2005)

    MathSciNet  MATH  Google Scholar 

  38. V. Totik, T. Varga, Chebyshev and fast decreasing polynomials. Proc. Lond. Math. Soc. (2015). doi:10.1112/plms/pdv014

    MATH  Google Scholar 

  39. V. Totik, P. Yuditskii, On a conjecture of Widom. J. Approx. Theory 190, 50–61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. W. Van Assche, Asymptotics for Orthogonal Polynomials. Lecture Notes in Mathematics, vol. 1265 (Springer, Berlin, 1987)

    Google Scholar 

  41. H. Widom, Polynomials associated with measures in the complex plane. J. Math. Mech. 16, 997–1013 (1967)

    MathSciNet  MATH  Google Scholar 

  42. H. Widom, Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first two authors are partially supported by TÜBİTAK (Scientific and Technological Research Council of Turkey), Project 115F199.

Author information

Authors and Affiliations

  1. Department of Mathematics, Bilkent University, Ankara, Turkey

    Gökalp Alpan & Alexander Goncharov

  2. Texas A&M University, College Station, TX, USA

    Burak Hatinoğlu

Authors
  1. Gökalp Alpan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Goncharov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Burak Hatinoğlu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Goncharov .

Editor information

Editors and Affiliations

  1. Dept of Mathematical Sciences, University of Memphis, Memphis, Tennessee, USA

    George A. Anastassiou

  2. Department of Mathematics, TOBB Univ. of Economics and Technology, Ankara, Ankara, Turkey

    Oktay Duman

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Alpan, G., Goncharov, A., Hatinoğlu, B. (2016). Some Asymptotics for Extremal Polynomials. In: Anastassiou, G., Duman, O. (eds) Computational Analysis. Springer Proceedings in Mathematics & Statistics, vol 155. Springer, Cham. https://doi.org/10.1007/978-3-319-28443-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation