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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0265832-2
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 11 (1956), no. 4 (70), 3-43; English transl., Amer. Math. Soc. Transl. (2) 22 (1962), 95-137. MR 18, 802. MR 0084064 (18:802f)
  • [2] M. M. Džrbašyan, Some questions of the theory of weighted polynomial approximations in a complex domain, Mat. Sb. 36 (78) (1955), 353-440. (Russian) MR 17, 31. MR 0070755 (17:31d)
  • [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 approximation by polynomials, Uspehi Mat. Nauk 11 (1956), no. 5 (71), 107-152; English transl., Amer. Math. Soc. Transl. (2) 10 (1958), 59-106. MR 18, 734; MR 20 #1146. MR 0083614 (18:734c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0265832-2
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

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