On polynomial approximation in the complex plane with application to conformal mapping
Author:
Lothar Reichel
Journal:
Math. Comp. 44 (1985), 425433
MSC:
Primary 30E10; Secondary 30C30, 41A10
MathSciNet review:
777274
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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 illconditioned, we show that a basis of Lagrange polynomials with Fejér points as nodes is wellconditioned. Numerical examples, where we compute polynomial approximations of conformal mappings, conclude the paper.
 [1]
M.
O. Afolabi and K.
O. Geddes, Nearminimax 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
(80g:65009)
 [2]
J.
H. Curtiss, Convergence of complex Lagrange interpolation
polynomials on the locus of the interpolation points, Duke Math. J.
32 (1965), 187–204. MR 0210902
(35 #1787)
 [3]
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
(80b:30007)
 [4]
G. H. Elliott, The Construction of Chebyshev Approximation in the Complex Plane, Thesis, University of London, 1979.
 [5]
Dieter
Gaier, Integralgleichungen erster Art und konforme Abbildung,
Math. Z. 147 (1976), no. 2, 113–129. MR 0396926
(53 #786)
 [6]
Dieter
Gaier, Vorlesungen über Approximation im Komplexen,
Birkhäuser Verlag, BaselBoston, Mass., 1980 (German). MR 604011
(82i:30055)
 [7]
Walter
Gautschi, The condition of orthogonal
polynomials, Math. Comp. 26 (1972), 923–924. MR 0313558
(47 #2112), http://dx.doi.org/10.1090/S00255718197203135589
 [8]
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 YorkLondon,
1978, pp. 45–72. MR 519056
(80b:65035)
 [9]
Walter
Gautschi, The condition of polynomials in power
form, Math. Comp. 33
(1979), no. 145, 343–352. MR 514830
(80f:65034), http://dx.doi.org/10.1090/S00255718197905148306
 [10]
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
(54 #10611)
 [11]
K.
O. Geddes and J.
C. Mason, Polynomial approximation by projections on the unit
circle, SIAM J. Numer. Anal. 12 (1975),
111–120. MR 0364977
(51 #1230)
 [12]
H.
Kober, Dictionary of conformal representations, Dover
Publications, Inc., New York, N. Y., 1952. MR 0049326
(14,156d)
 [13]
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
(80h:30008)
 [14]
J.
C. Mason, Recent advances in nearbest approximation,
Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980),
Academic Press, New YorkLondon, 1980, pp. 629–636. MR 602779
(82b:41025)
 [15]
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
(86h:65184), http://dx.doi.org/10.1016/03770427(84)900190
 [16]
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. 12, 125–142. Special
issue on numerical conformal mapping. MR 829034
(87h:65223), http://dx.doi.org/10.1016/03770427(86)901342
 [1]
 M. O. Afolabi & K. O. Geddes, "Nearminimax approximation of analytic functions on regular polygons," Proc. 7th Manitoba Conf. on Numer. Math. and Comp., University of Manitoba, Winnipeg, 1977, pp. 163176. MR 535008 (80g:65009)
 [2]
 J. H. Curtiss, "Convergence of complex Lagrange interpolation polynomials on the locus of the interpolation points," Duke Math. J., v. 32, 1965, pp. 187204. MR 0210902 (35:1787)
 [3]
 S. W. Ellacott, "A technique for approximate conformal mapping," in Multivariable Approximation (D. C. Handscomb, ed.), Academic Press, London, 1978. MR 525882 (80b:30007)
 [4]
 G. H. Elliott, The Construction of Chebyshev Approximation in the Complex Plane, Thesis, University of London, 1979.
 [5]
 D. Gaier, "Integralgleichungen erster Art und konforme Abbildung," Math. Z., v. 147, 1976, pp. 113129. MR 0396926 (53:786)
 [6]
 D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser, Basel, 1980. MR 604011 (82i:30055)
 [7]
 W. Gautschi, "The condition of orthogonal polynomials," Math. Comp., v. 26, 1977, pp. 923924. MR 0313558 (47:2112)
 [8]
 W. Gautschi, "Questions of numerical condition related to polynomials," in Recent Advances in Numercial Analysis (C. de Boor and G. Golub, eds.), Academic Press, New York, 1978. MR 519056 (80b:65035)
 [9]
 W. Gautschi, "Condition of polynomials in power form," Math. Comp., v. 33, 1979, pp. 343352. MR 514830 (80f:65034)
 [10]
 K. O. Geddes, "Chebyshev nodes for interpolation on a class of ellipses," in Theory of Approximation with Applications (A. G. Law and B. N. Sahney, eds.), Academic Press, New York, 1976. MR 0422625 (54:10611)
 [11]
 K. O. Geddes & J. C. Mason, "Polynomial approximation by projections on the unit circle," SIAM. J. Numer. Anal., v. 12, 1975, pp. 111120. MR 0364977 (51:1230)
 [12]
 H. Kober, Dictionary of Conformal Representations, Dover, New York, 1957. MR 0049326 (14:156d)
 [13]
 D. Levin, N. Papamichael & A. Sideridis, "The Bergman kernel method for the numerical conformai mapping of simply connected domains," J. Inst. Math. Appl., v. 22, 1978, pp. 171187. MR 509155 (80h:30008)
 [14]
 J. C. Mason, "Recent advances in nearbest approximation," in Approximation Theory III (E. W. Cheney, ed.), Academic Press, New York, 1980. MR 602779 (82b:41025)
 [15]
 L. Reichel, "On the determination of boundary collocation points for some problems for the Laplace operator," J. Comput. Appl. Math., v. 11, 1984, pp. 175196. MR 765969 (86h:65184)
 [16]
 L. Reichel, "A fast method for solving certain integral equations of the first kind with application to conformal mapping," J. Comput. Appl. Math. (To appear.) MR 829034 (87h:65223)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507772740
PII:
S 00255718(1985)07772740
Keywords:
Polynomial approximation,
polynomial basis,
numerical condition,
conformal mapping
Article copyright:
© Copyright 1985
American Mathematical Society
