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Properties and Examples of Faber–Walsh Polynomials

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Abstract

The Faber–Walsh polynomials are a direct generalization of the (classical) Faber polynomials from simply connected sets to sets with several simply connected components. In this paper, we derive new properties of the Faber–Walsh polynomials, where we focus on results of interest in numerical linear algebra, and on the relation between the Faber–Walsh polynomials and the classical Faber and Chebyshev polynomials. Moreover, we present examples of Faber–Walsh polynomials for two real intervals as well as for some non-real sets consisting of several simply connected components.

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References

  1. Achieser, N.I.: Theory of approximation. Charles J. Hyman. Frederick Ungar Publishing Co., New York (1956)

    MATH  Google Scholar 

  2. Akhiezer, N.: Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen. I. Bull. Acad. Sci. URSS 1932(9), 1163–1202 (1932)

    MATH  Google Scholar 

  3. Akhiezer, N.: Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen. II. Bull. Acad. Sci. URSS 1933(3), 309–344 (1933)

    MATH  Google Scholar 

  4. Beckermann, B., Reichel, L.: Error estimates and evaluation of matrix functions via the Faber transform. SIAM J. Numer. Anal. 47(5), 3849–3883 (2009). doi:10.1137/080741744

  5. Curtiss, J.H.: Faber polynomials and the Faber series. Am. Math. Mon. 78, 577–596 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  6. Driscoll, T.A., Toh, K.C., Trefethen, L.N.: From potential theory to matrix iterations in six steps. SIAM Rev. 40(3), 547–578 (1998). doi:10.1137/S0036144596305582

  7. Eiermann, M.: On semiiterative methods generated by Faber polynomials. Numer. Math. 56(2-3), 139–156 (1989). doi:10.1007/BF01409782

  8. Eiermann, M., Li, X., Varga, R.S.: On hybrid semi-iterative methods. SIAM J. Numer. Anal. 26(1), 152–168 (1989). doi:10.1137/0726009

  9. Eiermann, M., Niethammer, W.: On the construction of semi-iterative methods. SIAM J. Numer. Anal. 20(6), 1153–1160 (1983). doi:10.1137/0720085

  10. Eiermann, M., Niethammer, W., Varga, R.S.: A study of semi-iterative methods for nonsymmetric systems of linear equations. Numer. Math. 47(4), 505–533 (1985). doi:10.1007/BF01389454

  11. Ellacott, S.W.: Computation of Faber series with application to numerical polynomial approximation in the complex plane. Math. Comp. 40(162), 575–587 (1983). doi:10.2307/2007534

  12. Ellacott, S.W.: A survey of Faber methods in numerical approximation. Comput. Math. Appl. Part B 12(5–6), 1103–1107 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Faber, G.: Über polynomische Entwickelungen. Math. Ann. 57(3), 389–408 (1903). doi:10.1007/BF01444293

  14. Fischer, B.: Polynomial based iteration methods for symmetric linear systems. Wiley-Teubner Series Advances in Numerical Mathematics. Wiley, Chichester, B. G. Teubner, Stuttgart (1996). doi:10.1007/978-3-663-11108-5

  15. Fischer, B., Peherstorfer, F.: Chebyshev approximation via polynomial mappings and the convergence behaviour of Krylov subspace methods. Electron. Trans. Numer. Anal. 12, 205–215 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Grunsky, H.: Über konforme Abbildungen, die gewisse Gebietsfunktionen in elementare Funktionen transformieren. I. Math. Z. 67, 129–132 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grunsky, H.: Über konforme Abbildungen, die gewisse Gebietsfunktionen in elementare Funktionen transformieren. II. Math. Z. 67, 223–228 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grunsky, H.: Lectures on theory of functions in multiply connected domains. Vandenhoeck & Ruprecht, Göttingen (1978)

    MATH  Google Scholar 

  19. Hasson, M.: The capacity of some sets in the complex plane. Bull. Belg. Math. Soc. Simon Stevin 10(3), 421–436 (2003)

    MathSciNet  MATH  Google Scholar 

  20. Hasson, M.: The degree of approximation by polynomials on some disjoint intervals in the complex plane. J. Approx. Theory 144(1), 119–132 (2007). doi:10.1016/j.jat.2006.05.003

  21. Heuveline, V., Sadkane, M.: Arnoldi-Faber method for large non-Hermitian eigenvalue problems. Electron. Trans. Numer. Anal. 5, 62–76 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Jenkins, J.A.: On a canonical conformal mapping of J. L. Walsh. Trans. Am. Math. Soc. 88, 207–213 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kamo, S., Borodin, P.: Chebyshev polynomials for Julia sets. Mosc. Univ. Math. Bull. 49(5), 44–45 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Koch, T., Liesen, J.: The conformal “bratwurst” maps and associated Faber polynomials. Numer. Math. 86(1), 173–191 (2000). doi:10.1007/PL00005401

  25. Landau, H.J.: On canonical conformal maps of multiply connected domains. Trans. Am. Math. Soc. 99, 1–20 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liesen, J.: On the location of the zeros of Faber polynomials. Analysis (Munich) 20(2), 157–162 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Liesen, J.: Faber polynomials corresponding to rational exterior mapping functions. Constr. Approx. 17(2), 267–274 (2001). doi:10.1007/s003650010021

  28. Markushevich, A.I.: Theory of functions of a complex variable, vol. I. Prentice-Hall Inc., Englewood Cliffs (1965)

    MATH  Google Scholar 

  29. Markushevich, A.I.: Theory of functions of a complex variable, vol. III. Prentice-Hall Inc, Englewood Cliffs (1967)

    MATH  Google Scholar 

  30. Moret, I., Novati, P.: The computation of functions of matrices by truncated Faber series. Numer. Funct. Anal. Optim. 22(5-6), 697–719 (2001). doi:10.1081/NFA-100105314

  31. Moret, I., Novati, P.: An interpolatory approximation of the matrix exponential based on Faber polynomials. J. Comput. Appl. Math. 131(1-2), 361–380 (2001). doi:10.1016/S0377-0427(00)00261-2

  32. Nasser, M.M.S., Liesen, J., Sète, O.: Numerical computation of the conformal map onto lemniscatic domains. Comput. Methods Funct. Theory, 1–27 (2016). doi:10.1007/s40315-016-0159-x

  33. Peherstorfer, F.: Inverse images of polynomial mappings and polynomials orthogonal on them. In: Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), vol. 153, pp. 371–385 (2003). doi:10.1016/S0377-0427(02)00628-3

  34. Peherstorfer, F., Schiefermayr, K.: Description of extremal polynomials on several intervals and their computation. I, II. Acta Math. Hungar. 83(1–2), 27–58, 59–83 (1999). doi:10.1023/A:1006607401740

  35. Peherstorfer, F., Steinbauer, R.: Orthogonal and \(L_q\)-extremal polynomials on inverse images of polynomial mappings. J. Comput. Appl. Math. 127(1–2), 297–315 (2001). doi:10.1016/S0377-0427(00)00502-1

  36. Ransford, T.: Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28. Cambridge University Press, Cambridge (1995). doi:10.1017/CBO9780511623776

  37. Schiefermayr, K.: A lower bound for the minimum deviation of the Chebyshev polynomial on a compact real set. East J. Approx. 14(2), 223–233 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Schiefermayr, K.: Estimates for the asymptotic convergence factor of two intervals. J. Comput. Appl. Math. 236(1), 28–38 (2011). doi:10.1016/j.cam.2010.06.008

  39. Schiefermayr, K.: A lower bound for the norm of the minimal residual polynomial. Constr. Approx. 33(3), 425–432 (2011). doi:10.1007/s00365-010-9119-2

  40. Sète, O.: Some properties of Faber–Walsh polynomials (2013). arXiv:1306.1347v1

  41. Sète, O., Liesen, J.: On conformal maps from multiply connected domains onto lemniscatic domains. Electron. Trans. Numer. Anal. 45, 1–15 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Smirnov, V.I., Lebedev, N.A.: Functions of a complex variable: constructive theory. Russian by Scripta Technica Ltd. The M.I.T. Press, Cambridge, Mass (1968)

  43. Starke, G., Varga, R.S.: A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations. Numer. Math. 64(2), 213–240 (1993). doi:10.1007/BF01388688

  44. Suetin, P.K.: Series of faber polynomials, analytical methods and special functions, vol. 1. Gordon and Breach Science Publishers, Amsterdam (1998)

    Google Scholar 

  45. Walsh, J.L.: On the conformal mapping of multiply connected regions. Trans. Am. Math. Soc. 82, 128–146 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  46. Walsh, J.L.: A generalization of Faber’s polynomials. Math. Ann. 136, 23–33 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  47. Walsh, J.L.: Interpolation and approximation by rational functions in the complex domain, 5th edn. American Mathematical Society, Providence (1969)

    Google Scholar 

  48. Wegert, E.: Visual complex functions. Birkhäuser/Springer Basel AG, Basel (2012). doi:10.1007/978-3-0348-0180-5

  49. Wegert, E., Semmler, G.: Phase plots of complex functions: a journey in illustration. Notices Amer. Math. Soc. 58(6), 768–780 (2011)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank the anonymous referees for helpful comments.

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  1. TU Berlin, MA 4-5, Straße des 17. Juni 136, 10623, Berlin, Germany

    Olivier Sète & Jörg Liesen

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  1. Olivier Sète
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  2. Jörg Liesen
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Correspondence to Olivier Sète.

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Communicated by Lothar Reichel.

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Sète, O., Liesen, J. Properties and Examples of Faber–Walsh Polynomials. Comput. Methods Funct. Theory 17, 151–177 (2017). https://doi.org/10.1007/s40315-016-0176-9

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