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Mathematics of Computation

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The Faber polynomials for annular sectors

Authors: John P. Coleman and Nick J. Myers
Journal: Math. Comp. 64 (1995), 181-203, S1
MSC: Primary 30C10; Secondary 30C20, 30E10
MathSciNet review: 1260127
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Abstract: A conformal mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the Faber polynomials for that region. These polynomials are of interest in providing near-optimal polynomial approximations in a variety of contexts, including the construction of semiiterative methods for linear equations. The relevant conformal map for an annular sector $ \{ z:R \leq \vert z\vert \leq 1,\theta \leq \vert\arg z\vert \leq \pi \} $, with $ 0 < \theta \leq \pi $, is derived here and a recurrence relation is established for the coefficients of its Laurent expansion about the point at infinity. The recursive evaluation of scaled Faber polynomials is formulated in such a way that an algebraic manipulation package may be used to generate explicit expressions for their coefficients, in terms of two parameters which are determined by the interior angle of the annular sector and the ratio of its radii. Properties of the coefficients of the scaled Faber polynomials are established, and those for polynomials of degree $ \leq 15$ are tabulated in a Supplement at the end of this issue. A simple closed form is obtained for the coefficients of the Faber series for 1/z. Known results for an interval, a circular arc, and a circular sector are reproduced as special cases.

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Keywords: Faber polynomials, conformal mapping, annular sector, Faber series, transfinite diameter
Article copyright: © Copyright 1995 American Mathematical Society