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

Full-text PDF

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 , 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.

**[1]**J. P. Coleman, "Polynomial approximations in the complex plane,"*J. Comput. Appl. Math.*(In press.)**[2]**J. H. Curtiss, "Faber polynomials and Faber series,"*Amer. Math. Monthly*, v. 78, 1971, pp. 577-596. MR**0293104 (45:2183)****[3]**S. W. Ellacott, "Computation of Faber series with application to numerical polynomial approximation in the complex plane,"*Math. Comp.*, v. 40, 1983, pp. 575-587. MR**689474 (84e:65021)****[4]**G. H. Elliott,*The Construction of Chebyshev Approximations in the Complex Plane*, Ph.D. Thesis, University of London, 1978.**[5]**D. Gaier,*Vorlesungen über Approximation im Komplexen*, Birkhäuser, Basel, 1980. MR**604011 (82i:30055)****[6]**K. O. Geddes & J. C. Mason, "Polynomial approximation by projection on the unit circle,"*SIAM J. Numer. Anal.*, v. 12, 1975, pp. 111-120. MR**0364977 (51:1230)****[7]**U. Grothkopf & G. Opfer, "Complex Chebyshev polynomials on circular sectors with degree six or less,"*Math. Comp.*, v. 39, 1982, pp. 599-615. MR**669652 (84a:30070)****[8]**H. Kober,*Dictionary of Conformal Representations*, Dover, New York, 1957. MR**0049326 (14:156d)****[9]**J. N. Lyness & G. Sande, "Algorithm 413: Evaluation of normalized Taylor coefficients of an analytic function,"*Comm. ACM*, v. 14, 1971, pp. 669-675.**[10]**A. I. Markushevich,*Theory of Functions of a Complex Variable*, Chelsea, New York, 1977.**[11]**Ch. Pommerenke, "Über die Faberschen Polynome schlichter Funktionen,"*Math. Z.*, v. 85, 1964, pp. 197-208. MR**0168772 (29:6028)****[12]**V. I. Smirnov & N. A. Lebedev,*Functions of a Complex Variable, Constructive Theory*, Iliffe, London, 1968. MR**0229803 (37:5369)****[13]**J. L. Walsh,*Interpolation and Approximation by Rational Functions in the Complex Domain*, 5th ed., Amer. Math. Soc. Colloq. Publ., Vol. 20, 1969. MR**0218588 (36:1672b)**

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

Article copyright:
© Copyright 1987
American Mathematical Society