Abstract
Approximation by rational functions Rn (z) (in the C and Lp metrics) on plane compacta is investigated. The possibility is studied of the coincidence of rational and polynomial approximations for all n, and some functions are described for which this coincidence holds. Approximations on finite sets of points are investigated, and an explanation is given of why there are functions which cannot be approximated by rational functions of degree not higher than n (in the C metric).
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A. L. Levin and V. M. Tikhomirov, “The approximation of analytic functions by rational functions,” Dokl. Akad. Nauk SSSR, 174, No. 2, 279–282 (1967).
V. S. Videnskii, “Uniform approximation in the complex plane,” Uspekhi Matem. Nauk,11, No. 5, 169–175 (1956).
A. D. Ioffe and V. M. Tikhomirov, “Duality of convex functions in extremal problems,” Uspekhi Matem. Nauk,23, No. 6, 95 (1968).
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Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 121–130, February, 1971.
Several questions concerning this work were discussed with the author by A. A. Gonchar. The author wishes to thank A. A. Gonchar for his help.
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Levin, A.L. Approximation by rational functions in complex regions. Mathematical Notes of the Academy of Sciences of the USSR 9, 72–77 (1971). https://doi.org/10.1007/BF01316983
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DOI: https://doi.org/10.1007/BF01316983