The Faber polynomials for circular sectors
Authors:
John P. Coleman and Russell A. Smith
Journal:
Math. Comp. 49 (1987), 231241, 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 , 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 halfangle , , , , , 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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 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. 577596. 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. 575587. 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. 111120. 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. 599615. 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. 669675.
 [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. 197208. 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)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708902644
PII:
S 00255718(1987)08902644
Article copyright:
© Copyright 1987
American Mathematical Society
