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Mathematics of Computation

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The Faber polynomials for circular sectors

Authors: John P. Coleman and Russell A. Smith
Journal: Math. Comp. 49 (1987), 231-241, S1
MSC: Primary 30C30; Secondary 30E10, 65D20, 65E05
MathSciNet review: 890264
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Abstract: The Faber polynomials for a region of the complex plane, which are of interest as a basis for polynomial approximation of analytic functions, are determined by a conformal mapping of the complement of that region to the complement of the unit disc. We derive this conformal mapping for a circular sector $ \{ {z:\vert z\vert\; \leqslant 1,\;\vert\arg z\vert\; \leqslant \pi /\alpha } \}$, where $ \alpha > 1$, and obtain a recurrence relation for the coefficients of its Laurent expansion about the point at infinity. We discuss the computation of the coefficients of the Faber polynomials of degree 1 to 15, which are tabulated here for sectors of half-angle $ {5^ \circ }$, $ {10^\circ }$, $ {15^ \circ }$, $ {30^ \circ }$, $ {45^\circ }$, and $ {90^ \circ }$, and we give explicit expressions, in terms of $ \alpha $, for the polynomials of degree $ \leqslant 3$. The norms of Faber polynomials are tabulated and are compared with those of the Chebyshev polynomials for the same regions.

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Article copyright: © Copyright 1987 American Mathematical Society

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