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Faber polynomials corresponding to rational exterior mapping functions

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Abstract

Faber polynomials corresponding to rational exterior mapping functions of degree (m, m − 1) are studied. It is shown that these polynomials always satisfy an (m + 1)-term recurrence. For the special case m = 2, it is shown that the Faber polynomials can be expressed in terms of the classical Chebyshev polynomials of the first kind. In this case, explicit formulas for the Faber polynomials are derived.

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References

  1. J. P. Coleman, N. J. Myers (1995): The Faber polynomials for annular sectors. Math. Comp., 64:181–203.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. P. Coleman, R. A. Smith (1987): The Faber polynomials for circular sectors. Math. Comp., 49:231–241.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Curtiss (1971): Faber polynomials and the Faber series. Amer. Math. Monthly, 78:577–596.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Eiermann (1989): On semi-iterative methods generated by Faber polynomials. Numer. Math., 56:139–156.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Eiermann, R. S. Varga (1993): Zeros and local extreme points of Faber polynomials associated with hypocycloidal domains. Electron. Trans. Numer. Anal., 1:49–71 (electronic only).

    MATH  MathSciNet  Google Scholar 

  6. G. Faber (1903): Über polynomische Entwickelunger. Math. Ann., 57:398–408.

    Google Scholar 

  7. K. Gatermann, C. Hoffmann, G. Opfer (1992): Explicit Faber polynomials on circular sectors. Math. Comp., 58:241–253.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. He (1994): The Faber polynomials for circular arcs. In: Mathematics of Computation 1943–1993: A Half-Centry of Computational Mathematics (Vancouver, BC, 1993). Proc. Sympos. Appl. Math., Vol. 48, pp. 301–304, Providence, RI: American Mathematical Society.

    Google Scholar 

  9. M. He (1995): The Faber polynomials for circular lunes. Comput. Math. Appl., 30:307–315.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. X. He (1996): Explicit representations of Faber polynomials for m-cusped hypocycloids. J. Approx. Theory, 87:137–147.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. X. He, E. B. Saff (1994): The zeros of Faber polynomials for an m-cusped hypocycloid. J. Approx. Theory, 78:410–432.

    Article  MATH  MathSciNet  Google Scholar 

  12. T. Koch, J. Liesen (2000): The conformai “bratwurst” maps and associated Faber polynomials. Pub-lished online in Numerische Mathematik, DOI: 10.1007/s002110000141.

  13. J. Liesen (1998): Construction and analysis of polynomial iterative methods for non-Hermitian systems of linear equations. PhD thesis, Fakultät für Mathematik, Universität Bielefeld, http://archiv.ub.uni-bielefeld.de/disshabi/mathe.htm.

  14. T. A. Manteuffel, G. Starke, R. S. Varga (1995): Adaptive K-step iterative methods for nonsymmetric systems of linear equations. Electron. Trans. Numer. Anal., 3:50–65 (electronic).

    MATH  MathSciNet  Google Scholar 

  15. V. Smirnov, N. Lebedev (1968): Functions of a complex variable—Constructive theory. Cambridge, MA: MIT Press.

    Google Scholar 

  16. P. K. Suetin (1998): Series of Faber Polynomials. Amsterdam: Gordon and Breach.

    MATH  Google Scholar 

  17. L. N. Trefethen (1981): Rational Chebyshev approximation on the unit disk. Numer. Math., 37:297–320.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Fakultät für Mathematik, Universität Bielefeld, 33501, Bielefeld, Germany

    J. Liesen

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  1. J. Liesen
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Communicated by Dieter Gaier.

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Liesen, J. Faber polynomials corresponding to rational exterior mapping functions. Constr. Approx 17, 267–274 (2001). https://doi.org/10.1007/s003650010021

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  • DOI: https://doi.org/10.1007/s003650010021

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