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

DOI:
https://doi.org/10.1090/S0025-5718-1987-0890264-4

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 , where , 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 , , , , , and , and we give explicit expressions, in terms of , for the polynomials of degree . The norms of Faber polynomials are tabulated and are compared with those of the Chebyshev polynomials for the same regions.

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0890264-4

Article copyright:
© Copyright 1987
American Mathematical Society