Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 30C30, 30E10, 65D20, 65E05

Retrieve articles in all journals with MSC: 30C30, 30E10, 65D20, 65E05


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1987-0890264-4
PII: S 0025-5718(1987)0890264-4
Article copyright: © Copyright 1987 American Mathematical Society