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 , a closed Jordan curve. Lower bounds for the uniform norm of the difference on , where p is any polynomial, are obtained (in terms of the norm of R on ). In some cases these bounds are independent of as well as R and p. Some related results are also given.
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-  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.
-  Antoni Zygmund, Trigonometrical series, Chelsea Publishing Co., New York, 1952. 2nd ed. MR 0076084
- S. J. Poreda and G. S. Shapiro, Lower bounds for polynomial approximations to rational functions, Rocky Mountain J. Math. (to appear). MR 0361095 (50:13541)
- 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.
- A. Zygmund, Trigonometrical series, 2nd ed. reprinted with corrections and some additions, Cambridge Univ. Press, New York, 1968. MR 38 #4882. MR 0076084 (17:844d)
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Keywords: Rational function, polynomial, closed Jordan curve
Article copyright: © Copyright 1974 American Mathematical Society