A new method for Chebyshev approximation of complex-valued functions

Authors:
K. Glashoff and K. Roleff

Journal:
Math. Comp. **36** (1981), 233-239

MSC:
Primary 65D15; Secondary 30E10, 90C30

DOI:
https://doi.org/10.1090/S0025-5718-1981-0595055-4

MathSciNet review:
595055

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we are concerned with a formulation of the Chebyshev approximation problem in the complex plane as a problem of linear optimization in the presence of infinitely many constraints. It is shown that there exist stable and fast algorithms for the solution of optimization problems of this type. Some numerical examples are presented.

**[1]**D. O. Andreassen & G. A. Watson, "Linear Chebyshev approximation without Chebyshev sets,"*BIT*, v. 16, 1976, pp. 349-362. MR**0451625 (56:9907)****[2]**I. Barrodale, L. M. Delves & J. C. Mason, "Linear Chebyshev approximation of complexvalued functions,"*Math. Comp.*, v. 32, 1978, pp. 853-863. MR**0483298 (58:3313)****[3]**R. H. Bartels & G. H. Golub, "The simplex method of linear programming using*LU*-decompositions,"*Comm. ACM*, v. 12, 1969, pp. 266-268.**[4]**C. Carasso,*L'Algorithme d'Echange en Optimisation Convexe*, These, Grenoble, 1973.**[5]**A. Charnes, W. W. Cooper & K. O. Kortanek, "Duality, Haar programs and finite sequence spaces,"*Proc. Nat. Acad. Sci. U. S. A.*, v. 48, 1962, pp. 783-786. MR**0186393 (32:3853)****[6]**S. Ellacott & J. Williams, "Rational Chebyshev approximation in the complex plane,"*SIAM J. Numer. Anal.*, v. 13, 1976, pp. 310-323. MR**0413449 (54:1563)****[7]**K. Fahlander,*Computer Programs for Semi-Infinite Optimization*, TRITA-NA-7312, Department of Numerical Analysis, Royal Institute of Technology, S-10044, Stockholm 70, Sweden, 1973.**[8]**K. Glashoff & S. Å. Gustafson,*Einführung in die lineare Optimierung*, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978.**[9]**S. Å. Gustafson, "On the computational solution of a class of generalized moment problems,"*SIAM J. Numer. Anal.*, v. 7, 1970, pp. 343-357. MR**0270536 (42:5424)****[10]**S. Å. Gustafson, "Nonlinear system in semi-infinite programming," in*Numerical Solution of Nonlinear Algebraic Systems*(G. B. Byrnes & C. A. Hall, Eds.), Academic Press, New York, 1973, pp. 63-99.**[11]**K.-H. Hoffmann & A. Klostermair, "A semi-infinite linear programming procedure and applications to approximation problems in optimal control,"*Approximation Theory*II, Proc. Internat. Sympos., Austin, Texas, 1976, pp. 379-389. MR**0434406 (55:7372)****[12]**W. Krabs & G. Opfer, "Eine Methode zur Lösung des komplexen Approximationsproblems mit einer Anwendung auf konforme Abbildungen,"*Z. Angew. Math. Mech.*, v. 55, 1975, pp. 208-211. MR**0407495 (53:11270)****[13]**G. Opfer, "An algorithm for the construction of best approximations based on Kolmogorov's criterion,"*J. Approx. Theory*, v. 23, 1978, pp. 299-317. MR**509560 (80i:65019)**

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0595055-4

Article copyright:
© Copyright 1981
American Mathematical Society