Explicit Faber polynomials on circular sectors

Authors:
Karin Gatermann, Christoph Hoffmann and Gerhard Opfer

Journal:
Math. Comp. **58** (1992), 241-253, 51

MSC:
Primary 30C10; Secondary 30-04, 30C20, 65A05

DOI:
https://doi.org/10.1090/S0025-5718-1992-1106967-4

MathSciNet review:
1106967

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Abstract: We present explicit and precise expressions for the Faber polynomials on circular sectors up to degree 20. The precision is obtained by modifying (and simultaneously speeding up) an algorithm of Coleman and Smith so that an essential part of the Faber polynomials can be represented using only rational numbers. The growth of the coefficients of the Faber polynomials is determined. In addition, explicit expressions are given for the area two-norm and line two-norm of these polynomials. A conjecture is stated with respect to the uniform (infinity) norm which would also allow the explicit computation of the corresponding uniform norms of the Faber polynomials. Apart from a table of Faber polynomials, there are several other tables and graphs that illustrate the behavior of the Faber polynomials.

**[1]**John P. Coleman and Russel A. Smith,*The Faber polynomials for circular sectors*, Math. Comp.**49**(1987), 231-241. MR**890264 (88e:30022)****[2]**-,*Supplement to the Faber polynomials for circular sectors*, Math. Comp.**49**(1987), S1-S4. MR**890264 (88e:30022)****[3]**J. H. Curtiss,*Faber polynomials and the Faber series*, Amer. Math. Monthly**78**(1971), 577-596. MR**0293104 (45:2183)****[4]**S. W. Ellacott,*Computation of Faber series with application to numerical polynomial approximation in the complex plane*, Math. Comp.**40**(1983), 575-587. MR**689474 (84e:65021)****[5]**-,*On Faber polynomials and Chebyshev polynomials*, Approximation Theory (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), Academic Press, 1983, pp. 457-464. MR**754376 (85h:30006)****[6]**Dieter Gaier,*Konstruktive Methoden der konformen Abbildung*, Springer, Berlin, Göttingen, Heidelberg, 1964. MR**0199360 (33:7507)****[7]**-,*Lectures on complex approximation*, Birkhäuser, Boston, Basel, Stuttgart, 1987. MR**894920 (88i:30059b)****[8]**Uwe Grothkopf and Gerhard Opfer,*Complex Chebyshev polynomials on circular sectors with degree six or less*, Math. Comp.**33**(1982), 111-120. MR**669652 (84a:30070)****[9]**T. Kövari and Christian Pommerenke,*On Faber polynomials and Faber expansions*, Math. Z.**99**(1967), 193-206. MR**0227429 (37:3013)****[10]**Christian Pommerenke,*Über die Faberschen Polynome schlichter Funktionen*, Math. Z.**85**(1964), 197-208. MR**0168772 (29:6028)****[11]**Gerhard Rayna,*REDUCE, software for algebraic computation*, Springer, New York, 1987.**[12]**Joseph L. Walsh,*Interpolation and approximation by rational functions in the complex domain*, 5th ed., Amer. Math. Soc., Providence, RI, 1969. MR**0218588 (36:1672b)****[13]**Stephen Wolfram,*Mathematica--a system for doing mathematics by computer*, Addison-Wesley, Redwood City, CA, 1988.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1992-1106967-4

Keywords:
Faber polynomials,
circular sectors,
outer conformal mapping of sectors

Article copyright:
© Copyright 1992
American Mathematical Society