Constructive Approximation

, Volume 29, Issue 3, pp 421–448 | Cite as

On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Article

Abstract

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial Fn(z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)]n. Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of Ω, we obtain asymptotic formulas for Fn that yield fine results on the limiting distribution of the zeros of Faber polynomials.

Keywords

Faber polynomials Asymptotic behavior Zeros of polynomials Equilibrium measure Schwarz reflection principle Conformal map 

Mathematics Subject Classification (2000)

30E10 30E15 30C10 30C15 

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References

  1. 1.
    Bartolomeo, J., He, M.: On Faber polynomials generated by an m-star. Math. Comput. 62, 277–287 (1994) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cassels, J.W.S.: An Introduction to Diophantine Approximation. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45. Cambridge University Press, New York (1957) MATHGoogle Scholar
  3. 3.
    Coleman, J.P., Myers, N.J.: The Faber polynomials for annular sectors. Math. Comput. 64, 181–203 (1995) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Coleman, J.P., Smith, R.A.: The Faber polynomials for circular sectors. Math. Comput. 49, 231–241 (1987) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Davis, P.J.: The Schwarz Function and Its Applications. The Mathematical Association of America, The Carus Mathematical Monographs, No. 17. Buffalo, NY (1974) Google Scholar
  6. 6.
    Gaier, D.: On the decrease of Faber polynomials in domains with piecewise analytic boundary. Analysis 21, 219–229 (2001) MATHMathSciNetGoogle Scholar
  7. 7.
    He, M.: The Faber polynomials for m-fold symmetric domains. J. Comput. Appl. Math. 54, 313–324 (1994) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    He, M.: The Faber polynomials for circular lunes. Comput. Math. Appl. 30, 307–315 (1995) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Knopp, K.: Theory of Functions, vol. 2. Dover, New York (1952) MATHGoogle Scholar
  10. 10.
    Kuijlaars, A.B.J.: The zeros of Faber polynomials generated by an m-star. Math. Comput. 65, 151–156 (1996) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kuijlaars, A.B.J., Saff, E.B.: Asymptotic distribution of the zeros of Faber polynomials. Math. Proc. Camb. Phil. Soc. 118, 437–447 (1995) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lehman, R.S.: Development of the mapping function at an analytic corner. Pac. J. Math. 7, 1437–1449 (1957) MATHMathSciNetGoogle Scholar
  13. 13.
    Martínez-Finkelshtein, A., McLaughlin, K.T.-R., Saff, E.B.: Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics. Constr. Approx. 24, 319–363 (2006) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Martínez-Finkelshtein, A., McLaughlin, K.T.-R., Saff, E.B. Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle. Int. Math. Res. Not. doi:10.1155/IMRN/2006/91426 (2006). Papers vol. 2006, article ID 91426, 43 pages Google Scholar
  15. 15.
    McLaughlin, K.T.-R., Miller, P.D.: The \(\overline{\partial}\) steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights. Int. Math. Res. doi:10.1155/IMRP/2006/48673 (2006). Papers vol. 2006, article ID 48673, 78 pages Google Scholar
  16. 16.
    Pritsker, I.E.: On the local asymptotics of Faber polynomials. Proc. Am. Math. Soc. 127, 2953–2960 (1999) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997) MATHGoogle Scholar
  18. 18.
    Simon, B.: Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures. ETNA 25, 328–368 (2006) MATHGoogle Scholar
  19. 19.
    Suetin, P.K.: Series of Faber Polynomials. Analytical Methods and Special Functions, vol. 1. Gordon and Breach Science, Amsterdam (1998) MATHGoogle Scholar
  20. 20.
    Szabados, J.: On some problems connected with polynomials orthogonal on the complex unit circle. Act. Math. Sci. Hung. 33, 197–210 (1979) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ullman, J.L.: Studies in Faber polynomials I. Trans. Am. Math. Soc. 73, 515–528 (1960) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA

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