Approximation on compact subsets of \(\mathbb{R}\). (English) Zbl 1044.41005
Buhmann, Martin D. (ed.) et al., Advanced problems in constructive approximation. 3rd international Dortmund meeting on approximation theory (IDoMAT), Dortmund, Germany, August 20–24, 2001. Basel: Birkhäuser (ISBN 3-7643-6648-6/hbk). ISNM, Int. Ser. Numer. Math. 142, 263-274 (2003).
The Weierstrass approximation theorem can be deduced from the following formula and the binomial theorem:
\[
| x|= [1- (1- x^2)]^{{1\over 2}}\qquad (-1\leq x\leq 1).
\]
By using fast decreasing polynomials, the author proves a refined version of this result due to Jackson. For earlier work on this theme, see [K. G. Ivanov and V. Totik, Fast decreasing polynomials, Constructive Approximation 6, 1–20 (1990; Zbl 0682.41014)].
For the entire collection see [Zbl 1007.00035].
For the entire collection see [Zbl 1007.00035].
Reviewer: H. R. Dowson (Glasgow)
