Explicit Faber polynomials on circular sectors

Gatermann, Karin; Hoffmann, Christoph; Opfer, Gerhard

doi:10.1090/S0025-5718-1992-1106967-4

Explicit Faber polynomials on circular sectors
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by Karin Gatermann, Christoph Hoffmann and Gerhard Opfer PDF

Math. Comp. 58 (1992), 241-253 Request permission

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.

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