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Approximation of complex harmonic functions by complex harmonic splines


Authors: Han Lin Chen and Tron Hvaring
Journal: Math. Comp. 42 (1984), 151-164
MSC: Primary 30C30; Secondary 30E10, 41A15
DOI: https://doi.org/10.1090/S0025-5718-1984-0725990-8
MathSciNet review: 725990
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Abstract: In this paper, a class of complex harmonic spline functions (C.H.S.) are defined on the unit disc U. We use the C.H.S. to approximate the complex harmonic function on U, showing that C.H.S. may be represented by elementary functions. If the maximum step tends to zero and the mesh ratio is bounded, then C.H.S. converge uniformly to the interpolated function F on the closed disc Ū. If the interpolated function F is a conformal mapping, then the C.H.S. is a quasi-conformal mapping.


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  • [1] J. H. Ahlberg, H. N. Nilson & J. L. Walsh, "Complex cubic splines," Trans. Amer. Math. Soc., v. 129, 1967, pp. 391-413. MR 0217484 (36:573)
  • [2] J. H. Ahlberg, Splines in the Complex Plane, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. Wisconsin, Madison, Wis., 1969), Academic Press, New York, 1969, pp. 1-27. MR 0257614 (41:2264)
  • [3] Chen Han-Lin, Complex Harmonic Splines, Interpolation and Approximation on the Unit Circle, Part II, Tech. Report, Math. and Comp. No. 3/81, ISBN 82-7151-039-8.
  • [4] Chen Han-Lin, "Complex spline functions," Sci. Sinica, v. 24, No. 2, 1981, pp. 160-169. MR 626312 (82h:41013)
  • [5] Chen Han-Lin & Tron Hvaring, A New Method for the Approximation of Conformal Mapping on the Unit Circle, Tech. Report, Math. and Comp. No. 6/82, ISBN 82-7151-049-5.
  • [6] L. Fejér, "Über Interpolation," Göttingen Nachr, 1916, pp. 66-91.
  • [7] D. Gaier, "Integralgleichungen erster Art und konforme Abbildung," Math. Z., v. 47, 1976, pp. 113-129. MR 0396926 (53:786)
  • [8] M. A. Lawrentjew & B. W. Schabat, Methoden der komplexen Funktionentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 0209447 (35:345)
  • [9] L. Reichel, On Polynomial Approximation in the Complex Plane with Application to Conformal Mapping, TRITA-Na-8102, Dept. Comput. Sci., Royal Institute of Technology, Stockholm.
  • [10] L. Reichel, On the Determination of Boundary Collocation Points for Solving some Problems for the Laplace Operator, TRITA-NA-8006, Dept. Comput. Sci., Royal Institute of Technology, Stockholm.
  • [11] A. Sharma & A. Meir, "Degree of approximation of spline interpolation," J. Math. and Mech., v. 15, No. 1-6, 1966, pp. 759-767. MR 0194800 (33:3006)
  • [12] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. MR 0114894 (22:5712)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1984-0725990-8
Article copyright: © Copyright 1984 American Mathematical Society

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