On polynomial approximation in the complex plane with application to conformal mapping
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Abstract:
We consider the selection of polynomial bases for polynomial approximation of analytic functions on bounded simply connected regions in the complex plane. While a monomial basis may be very ill-conditioned, we show that a basis of Lagrange polynomials with Fejér points as nodes is well-conditioned. Numerical examples, where we compute polynomial approximations of conformal mappings, conclude the paper.References
- M. O. Afolabi and K. O. Geddes, Near-minimax interpolation of analytic functions on regular polygons, Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1977) Congress. Numer., XX, Utilitas Math., Winnipeg, Man., 1978, pp. 163–176. MR 535008
- J. H. Curtiss, Convergence of complex Lagrange interpolation polynomials on the locus of the interpolation points, Duke Math. J. 32 (1965), 187–204. MR 210902
- S. W. Ellacott, A technique for approximate conformal mapping, Multivariate approximation (Sympos., Univ. Durham, Durham, 1977) Academic Press, London-New York, 1978, pp. 301–314. MR 525882 G. H. Elliott, The Construction of Chebyshev Approximation in the Complex Plane, Thesis, University of London, 1979.
- Dieter Gaier, Integralgleichungen erster Art und konforme Abbildung, Math. Z. 147 (1976), no. 2, 113–129. MR 396926, DOI 10.1007/BF01164277
- Dieter Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser Verlag, Basel-Boston, Mass., 1980 (German). MR 604011
- Walter Gautschi, The condition of orthogonal polynomials, Math. Comp. 26 (1972), 923–924. MR 313558, DOI 10.1090/S0025-5718-1972-0313558-9
- Walter Gautschi, Questions of numerical conditions related to polynomials, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 45–72. MR 519056
- Walter Gautschi, The condition of polynomials in power form, Math. Comp. 33 (1979), no. 145, 343–352. MR 514830, DOI 10.1090/S0025-5718-1979-0514830-6
- K. O. Geddes, Chebyshev nodes for interpolation on a class of ellipses, Theory of approximation, with applications (Proc. Conf., Univ. Calgary, Calgary, Alta., 1975; dedicated to the memory of Eckard Schmidt), Academic Press, New York, 1976, pp. 155–170. MR 0422625
- 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
- H. Kober, Dictionary of conformal representations, Dover Publications, Inc., New York, N.Y., 1952. MR 0049326
- 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. C. Mason, Recent advances in near-best approximation, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 629–636. MR 602779
- Lothar Reichel, On the determination of boundary collocation points for solving some problems for the Laplace operator, J. Comput. Appl. Math. 11 (1984), no. 2, 175–196. MR 765969, DOI 10.1016/0377-0427(84)90019-0
- Lothar Reichel, A fast method for solving certain integral equations of the first kind with application to conformal mapping, J. Comput. Appl. Math. 14 (1986), no. 1-2, 125–142. Special issue on numerical conformal mapping. MR 829034, DOI 10.1016/0377-0427(86)90134-2
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp. 44 (1985), 425-433
- MSC: Primary 30E10; Secondary 30C30, 41A10
- DOI: https://doi.org/10.1090/S0025-5718-1985-0777274-0
- MathSciNet review: 777274