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Mathematics of Computation

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Computation of Faber series with application to numerical polynomial approximation in the complex plane

Author: S. W. Ellacott
Journal: Math. Comp. 40 (1983), 575-587
MSC: Primary 65D15; Secondary 30B50, 30E10, 41A10
MathSciNet review: 689474
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Abstract: Kövari and Pommerenke [19], and Elliott [8], have shown that the truncated Faber series gives a polynomial approximation which (for practical values of the degree of the polynomial) is very close to the best approximation. In this paper we discuss efficient Fast Fourier Transform (FFT) and recursive methods for the computation of Faber polynomials, and point out that the FFT method described by Geddes [13], for computing Chebyshev coefficients can be generalized to compute Faber coefficients. We also give a corrected bound for the norm of the Faber projection (that given in Elliott [8], being unfortunately slightly in error) and very briefly discuss a possible extension of the method to the case when the mapping function, which is required to compute the Faber series, is not known explicitly.

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Keywords: Faber series, near best approximation
Article copyright: © Copyright 1983 American Mathematical Society