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Convergence of Julia polynomials

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Abstract

We study the approximation of conformal mappings with the polynomials defined by Keldysh and Lavrentiev from an extremal problem considered by Julia. These polynomials converge uniformly on the closure of any Smirnov domain to the conformal mapping of this domain onto a disk. We prove estimates for the rate of such convergence on domains with piecewise analytic boundaries, expressed through the smallest exterior angles at the boundary.

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References

  1. L. V. Ahlfors,Two numerical methods in conformal mapping, inExperiments in the Computation of Conformal Maps, National Bureau of Standards Applied Mathematics Series, No. 42, U. S. Government Printing Office, Washington, D.C., 1955, pp. 45–52.

    Google Scholar 

  2. V. V. Andrievskii, V. I. Belyi and V. K. Dzjadyk,Conformal Invariants in Constructive Theory of Functions of a Complex Variable, World Federation Publishers, Atlanta, 1995.

    Google Scholar 

  3. V. V. Andrievskii and H.-P. Blatt,Discrepancy of Signed Measures and Polynomial Approximation, Springer-Verlag, New York, 2002.

    MATH  Google Scholar 

  4. V. V. Andrievskii and D. Gaier,Uniform convergence of Bieberbach polynomials in domains with piecewise quasianalytic boundary, Mitt. Math. Sem. Giessen211 (1992), 49–60.

    MathSciNet  Google Scholar 

  5. V. V. Andrievskii and I. E. Pritsker,Convergence of Bieberbach polynomials in domains with interior cusps, J. Analyse Math.82 (2000), 315–332.

    Article  MATH  MathSciNet  Google Scholar 

  6. H.-P. Blatt and E. B. Saff,Behavior of zeros of polynomials of near best approximation, J. Approx. Theory46 (1986), 323–344.

    Article  MATH  MathSciNet  Google Scholar 

  7. H.-P. Blatt, E. B. Saff and M. Simkani,Jentzsch-Szegö type theorems for the zeros of best approximants, J. London Math. Soc.38 (1988), 307–316.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. L. Duren,Theory of H p Spaces, Academic Press, New York, 1970.

    MATH  Google Scholar 

  9. D. Gaier,Konstruktive Methoden der konformen Abbildung, Springer-Verlag, Berlin, 1964.

    MATH  Google Scholar 

  10. D. Gaier,On the convergence of the Bieberbach polynomials in regions with piecewise analytic boundary, Arch. Math.58 (1992), 289–305.

    Article  MathSciNet  Google Scholar 

  11. D. Gaier,Polynomial approximation of conformal maps, Constr. Approx.14 (1998), 27–40.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Julia,Lecons sur la représentation conforme des aires simplement connexes, Gauthier-Villars, Paris, 1931.

    MATH  Google Scholar 

  13. M. V. Keldysh,On a class of extremal polynomials, Dokl. Akad. Nauk SSSR4 (1936), 163–166. (Russian)

    Google Scholar 

  14. M. V. Keldysh and M. A. Lavrentiev,On the theory of conformal mappings, Dokl. Akad. Nauk SSSR1 (1935), 85–87. (Russian)

    Google Scholar 

  15. M. V. Keldysh and M. A. Lavrentiev,Sur la représentation conforme des domaines limités par des courbes rectifiables, Ann. Sci. École Norm. Sup.54 (1937), 1–38.

    MATH  Google Scholar 

  16. R. S. Lehman,Development of the mapping function at an analytic corner, Pacific J. Math.7 (1957), 1437–1449.

    MATH  MathSciNet  Google Scholar 

  17. R. Näkki and B. Palka,Lipschitz conditions, b-arcwise connectedness and conformal mappings, J. Analyse Math.42 (1982/83), 38–50.

    MathSciNet  Google Scholar 

  18. Ch. Pommerenke,Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.

    MATH  Google Scholar 

  19. I. E. Pritsker,Comparing norms of polynomials in one and several variables, J. Math. Anal. Appl.216 (1997), 685–695.

    Article  MATH  MathSciNet  Google Scholar 

  20. I. E. Pritsker,Approximation of conformal mapping via the Szegö kernel method, Comput. Methods Funct. Theory3 (2003), 79–94.

    MATH  MathSciNet  Google Scholar 

  21. T. Ransford,Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995.

    MATH  Google Scholar 

  22. P. C. Rosenbloom and S. E. Warschawski,Approximation by polynomials, inLectures on Functions of a Complex Variable (W. Kaplan, ed.), University of Michigan Press, Ann Arbor, 1955, pp. 287–302.

    Google Scholar 

  23. V. I. Smirnov and N. A. Lebedev,Functions of a Complex Variable: Constructive Theory, MIT Press, Cambridge, 1968.

    MATH  Google Scholar 

  24. G. Szegö,Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1975.

    MATH  Google Scholar 

  25. J. L. Walsh,Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, RI, 1965.

    MATH  Google Scholar 

  26. S. E. Warschawski,Recent results in numerical methods of conformal mapping, inProceedings of Symposia in Applied Mathematics. Vol. VI. Numerical Analysis (J. H. Curtiss, ed.), McGraw-Hill, New York, 1956, pp. 219–250.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, 74078-1058, Stillwater, OK, USA

    Igor E. Pritsker

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  1. Igor E. Pritsker
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Correspondence to Igor E. Pritsker.

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Research supported in part by the National Security Agency under Grant No. MDA904-03-1-0081.

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Pritsker, I.E. Convergence of Julia polynomials. J. Anal. Math. 94, 343–361 (2004). https://doi.org/10.1007/BF02789053

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  • DOI: https://doi.org/10.1007/BF02789053

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