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Transactions of the American Mathematical Society

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Fundamental constants for rational functions


Authors: S. J. Poreda, E. B. Saff and G. S. Shapiro
Journal: Trans. Amer. Math. Soc. 189 (1974), 351-358
MSC: Primary 30A82
MathSciNet review: 0361096
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Abstract: Suppose R is a rational function with n poles all of which lie inside $ \Gamma $, a closed Jordan curve. Lower bounds for the uniform norm of the difference $ R - p$ on $ \Gamma $, where p is any polynomial, are obtained (in terms of the norm of R on $ \Gamma $). In some cases these bounds are independent of $ \Gamma $ as well as R and p. Some related results are also given.


References [Enhancements On Off] (What's this?)

  • [1] S. J. Poreda and G. S. Shapiro, Lower bounds for polynomial approximations to rational functions, Proceedings of the International Conference on Padé Approximants, Continued Fractions and Related Topics (Univ. Colorado, Boulder, Colo., 1972; dedicated to the memory of H. S. Wall), 1974, pp. 377–378. MR 0361095
  • [2] A. I. Markuševič, Theory of analytic functions, GITTL, Moscow, 1950; English transl., Theory of functions of a complex variable, Prentice-Hall, Englewood Cliffs, N. J., 1967. MR 12,87; MR 35 #6799.
  • [3] Antoni Zygmund, Trigonometrical series, Chelsea Publishing Co., New York, 1952. 2nd ed. MR 0076084

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0361096-8
Keywords: Rational function, polynomial, closed Jordan curve
Article copyright: © Copyright 1974 American Mathematical Society