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 | References | Similar Articles | Additional Information
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 $\exp ( - |x{|^\alpha }),\alpha \geqq 2$. This lower bound has the same order of magnitude as the upper bound produced previously by Džrbašyan.
- 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
- 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 Dunham Jackson, The theory of approximations, Amer. Math. Soc. Colloq. Publ., vol. 11, Amer. Math. Soc., Providence, R. I., 1930.
- 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 <!– MATH $( - \infty ,\infty )$ –> <IMG WIDTH="86" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img4.gif" ALT="$( - \infty ,\infty )$">,
degree of approximation on <!– MATH $( - \infty ,\infty )$ –> <IMG WIDTH="86" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$( - \infty ,\infty )$">,
Jackson’s theorem on <!– MATH $( - \infty ,\infty )$ –> <IMG WIDTH="86" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$( - \infty ,\infty )$">,
weighted approximation on <!– MATH $( - \infty ,\infty )$ –> <IMG WIDTH="86" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="$( - \infty ,\infty )$">
Article copyright:
© Copyright 1970
American Mathematical Society