Chebyshev Polynomials Associated with a System of Continua

Chapter
Part of the Fields Institute Communications book series (FIC, volume 81)

Abstract

We establish estimates from above for the uniform norm of the Chebyshev polynomials associated with a system of continua \(K \subset \mathbb {C}\) by constructing monic polynomials with small norms on K. The estimates are exact (up to a constant factor) in the case where K has a piecewise quasiconformal boundary and its complement \(\varOmega =\overline {\mathbb {C}} \setminus K\) has no outward pointing cusps.

Keywords

Chebyshev polynomials Equilibrium measure Green’s function Quasiconformal curve System of continua 

Msc codes:

30C10 30C20 30C62 30C85 31A15 31A20 

Notes

Acknowledgements

I would like to extend an enormous thank you to Vladimir Andrievskii and Fedor Nazarov for providing me with motivation, ideas, and all of their thoughtful and useful feedback.

References

  1. 1.
    L. V. Ahlfors: Complex Analysis 3rd ed. Singapore: McGraw-Hill (1979)Google Scholar
  2. 2.
    L. V. Ahlfors: Lectures on Quasiconformal Mappings. Princeton, N.J.: Van Nostrand (1966)Google Scholar
  3. 3.
    V. V. Andrievskii: Chebyshev polynomials on a system of continua. Constr. Approx. (2016)Google Scholar
  4. 4.
    V. V. Andrievskii: Polynomial Approximation of Polyharmonic Functions on a Complement of a John Domain. J. Approx. Theory, 190:116–132 (2015)Google Scholar
  5. 5.
    V. V. Andrievskii, H.-P. Blatt: Discrepancy of Signed Measure and Polynomial Approximation. Berlin/New York: Springer-Verlag (2002)Google Scholar
  6. 6.
    A. Goncharov, B. Hartinoğlu: Widom Factors. Potential Anal., 42:671–680 (2015)Google Scholar
  7. 7.
    O. Lehto, K. I. Virtanen: Quasiconformal Mappings in the Plane. New York: Springer-Verlag (1973)Google Scholar
  8. 8.
    Ch. Pommerenke: Boundary Behaviour of Conformal Mappings. Berlin/New York: Springer-Verlag (1992)Google Scholar
  9. 9.
    T. Ransford: Potential Theory in the Plane. Cambridge: Cambridge University Press. (1995)Google Scholar
  10. 10.
    V. I. Smirnov, N. A. Lebedev: Functions of a Complex Variable: Constructive Theory. Constr. Theory, Cambridge: M.I.T. (1968)Google Scholar
  11. 11.
    M. L. Sodin, P. M. Yuditskii: Functions Least Deviating from Zero on Closed Subsets of the Real Line. St. Petersburg Math. J., 4:201–249 (1993)Google Scholar
  12. 12.
    P. K. Suetin: Series of Faber Polynomials (in Russian). Naukova Dumka, Kiev (1998)Google Scholar
  13. 13.
    V. Totik, E. B. Saff: Logarithmic Potentials with External Fields. Berlin/New York: Springer-Verlag (1997)Google Scholar
  14. 14.
    V. Totik: Chebyshev Polynomials on a System of Curves. Journal D’Analyse Mathématique, 118:317–338 (2012)Google Scholar
  15. 15.
    V. Totik: Chebyshev Polynomials on Compact Sets. Potential Anal., 40:511–524 (2013)Google Scholar
  16. 16.
    V. Totik: Asymptotics of Christoffel Functions on Arcs and Curves. Adv. Math., 252:114–149 (2014)Google Scholar
  17. 17.
    V. Totik, T. Varga: Chebyshev and Fast Decreasing Polynomials. Proc. London Math. Soc. (2015). doi: 10.1112/plms/pdv014Google Scholar
  18. 18.
    H. Widom: Extremal Polynomials Associated with a System of Curves in the Complex Plane. Adv. Math., 3:127–232 (1969)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Kent State UniversityKentUSA

Personalised recommendations