Computation of Faber series with application to numerical polynomial approximation in the complex plane
HTML articles powered by AMS MathViewer
 by S. W. Ellacott PDF
 Math. Comp. 40 (1983), 575587 Request permission
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.References

M. Abramowitz & I. A. Stegun, Handbook of Mathematical Functions (9th Dover Printing based on the U. S. Government Printing Office edition), Dover, New York, 1972.
 I. Barrodale, L. M. Delves, and J. C. Mason, Linear Chebyshev approximation of complexvalued functions, Math. Comp. 32 (1978), no. 143, 853–863. MR 483298, DOI 10.1090/S0025571819780483298X
 H.P. Blatt, A general Remes algorithm in real or complex normed linear spaces, Multivariate approximation (Sympos., Univ. Durham, Durham, 1977) Academic Press, LondonNew York, 1978, pp. 145–153. MR 525872
 E. W. Cheney and K. H. Price, Minimal projections, Approximation Theory (Proc. Sympos., Lancaster, 1969) Academic Press, London, 1970, pp. 261–289. MR 0265842
 J. H. Curtiss, Faber polynomials and the Faber series, Amer. Math. Monthly 78 (1971), 577–596. MR 293104, DOI 10.2307/2316567
 S. Ellacott and Jack Williams, Linear Chebyshev approximation in the complex plane using Lawson’s algorithm, Math. Comput. 30 (1976), no. 133, 35–44. MR 0400652, DOI 10.1090/S00255718197604006520
 S. W. Ellacott, A technique for approximate conformal mapping, Multivariate approximation (Sympos., Univ. Durham, Durham, 1977) Academic Press, LondonNew York, 1978, pp. 301–314. MR 525882 G. H. Elliott, The Construction of Chebyshev Approximations in the Complex Plane, Ph.D. Thesis, Faculty of Science (Mathematics), University of London, 1978.
 L. Fox and I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, LondonNew YorkToronto, Ont., 1968. MR 0228149
 Dieter Gaier, Konstruktive Methoden der konformen Abbildung, Springer Tracts in Natural Philosophy, Vol. 3, SpringerVerlag, Berlin, 1964 (German). MR 0199360
 Dieter Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser Verlag, BaselBoston, Mass., 1980 (German). MR 604011
 K. O. Geddes and J. C. Mason, Polynomial approximation by projections on the unit circle, SIAM J. Numer. Anal. 12 (1975), 111–120. MR 364977, DOI 10.1137/0712011
 K. O. Geddes, Nearminimax polynomial approximation in an elliptical region, SIAM J. Numer. Anal. 15 (1978), no. 6, 1225–1233. MR 512695, DOI 10.1137/0715083
 K. Glashoff and K. Roleff, A new method for Chebyshev approximation of complexvalued functions, Math. Comp. 36 (1981), no. 153, 233–239. MR 595055, DOI 10.1090/S00255718198105950554
 E. Grassmann, Numerical experiments with a method of successive approximation for conformal mapping, Z. Angew. Math. Phys. 30 (1979), no. 6, 873–884 (English, with German summary). MR 559243, DOI 10.1007/BF01590486
 HansPeter Hoidn, Osculation methods for the conformal mapping of doubly connected regions, Z. Angew. Math. Phys. 33 (1982), no. 5, 640–652 (English, with German summary). MR 680932, DOI 10.1007/BF00944947
 Martin H. Gutknecht, Ein Abstiegsverfahren für nichtdiskrete TschebyscheffApproximationsprobleme, Numerische Methoden der Approximationstheorie, Band 4 (Meeting, Math. Forschungsinst., Oberwolfach, 1977) Internat. Schriftenreihe Numer. Math., vol. 42, Birkhäuser, BaselBoston, Mass., 1978, pp. 154–171 (German, with English summary). MR 527102
 Martin H. Gutknecht and Lloyd N. Trefethen, Real polynomial Chebyshev approximation by the CarathéodoryFejér method, SIAM J. Numer. Anal. 19 (1982), no. 2, 358–371. MR 650056, DOI 10.1137/0719022
 T. Kövari and Ch. Pommerenke, On Faber polynomials and Faber expansions, Math. Z. 99 (1967), 193–206. MR 227429, DOI 10.1007/BF01112450
 D. Levin, N. Papamichael, and A. Sideridis, The Bergman kernel method for the numerical conformal mapping of simply connected domains, J. Inst. Math. Appl. 22 (1978), no. 2, 171–187. MR 509155 J. N. Lyness & G. Sande, "Algorithm 413: Evaluation of normalised Taylor coefficients of an analytic function," Comm. ACM, v. 14, 1971, pp. 669675. A. I. Markushevich, Theory of Functions of a Complex Variable, Translated and edited by R. A. Silverman (three volumes in one), Chelsea, New York, 1977.
 L. M. Nepritvorennaja, Construction of Faber polynomials for an arbitrary polygon, Bul. Akad. Štiince RSS Moldoven. 3 (1973), 80–83, 93 (Russian). MR 333136
 J. C. Mason, Orthogonal polynomial approximation methods in numerical analysis, Approximation Theory (Proc. Sympos., Lancaster, 1969) Academic Press, London, 1970, pp. 7–33. MR 0270034
 Gerhard Opfer, An algorithm for the construction of best approximations based on Kolmogorov’s criterion, J. Approx. Theory 23 (1978), no. 4, 299–317 (English, with German summary). MR 509560, DOI 10.1016/00219045(78)900825
 Gerhard Opfer, An algorithm for the construction of best approximations based on Kolmogorov’s criterion, J. Approx. Theory 23 (1978), no. 4, 299–317 (English, with German summary). MR 509560, DOI 10.1016/00219045(78)900825
 Christian Pommerenke, Konforme Abbildung und FeketePunkte, Math. Z. 89 (1965), 422–438 (German). MR 181740, DOI 10.1007/BF01112271 J. Radon, "Über die Randwertaufgaben beim logarithmischen Potential," Sitz.Ber. Akad. Wiss., Wien. Math.naturw. Kl., Abt. IIa, v. 128, 1919, pp. 11231167. R. L. Streit & A. H. Nutall, Linear Chebyshev Complex Function Approximation, NUSC report 6403, Naval Underwater Systems Center, New London, Conn., 1981.
 Lloyd N. Trefethen, Numerical computation of the SchwarzChristoffel transformation, SIAM J. Sci. Statist. Comput. 1 (1980), no. 1, 82–102. MR 572542, DOI 10.1137/0901004
 Lloyd N. Trefethen, Computer application of the SchwarzChristoffel transformation, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, BaselBoston, Mass., 1981, pp. 263–274. MR 661071
 Lloyd N. Trefethen, Nearcircularity of the error curve in complex Chebyshev approximation, J. Approx. Theory 31 (1981), no. 4, 344–367. MR 628517, DOI 10.1016/00219045(81)901027
Additional Information
 © Copyright 1983 American Mathematical Society
 Journal: Math. Comp. 40 (1983), 575587
 MSC: Primary 65D15; Secondary 30B50, 30E10, 41A10
 DOI: https://doi.org/10.1090/S00255718198306894747
 MathSciNet review: 689474