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Proceedings of the American Mathematical Society

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A lower Jackson bound on $ (-\infty ,\,\infty )$

Authors: J. S. Byrnes and D. J. Newman
Journal: Proc. Amer. Math. Soc. 26 (1970), 71-72
MSC: Primary 41.41
MathSciNet review: 0265832
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Abstract: We produce a lower bound for the degree of uniform polynomial approximation to continuous functions on the whole real line using the weight function $ \exp ( - \vert x{\vert^\alpha }),\alpha \geqq 2$. This lower bound has the same order of magnitude as the upper bound produced previously by Džrbašyan.

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  • [1] N. I. Ahiezer, On weighted approximations of continuous functions by polynomials on the entire number axis, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 4(70), 3–43 (Russian). MR 0084064
  • [2] M. M. Džrbašyan, Some questions of the theory of weighted polynomial approximations in a complex domain, Mat. Sb. N.S. 36(78) (1955), 353–440 (Russian). MR 0070755
  • [3] Dunham Jackson, The theory of approximations, Amer. Math. Soc. Colloq. Publ., vol. 11, Amer. Math. Soc., Providence, R. I., 1930.
  • [4] S. N. Mergelyan, Weighted approximations by polynomials, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 5(71), 107–152 (Russian). MR 0083614

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Keywords: Polynomial approximation on $ ( - \infty ,\infty )$, degree of approximation on $ ( - \infty ,\infty )$, Jackson's theorem on $ ( - \infty ,\infty )$, weighted approximation on $ ( - \infty ,\infty )$
Article copyright: © Copyright 1970 American Mathematical Society