ISSN:
1432-0940
Keywords:
41A25
;
Best uniform approximation
;
Polynomials
;
Chebyshev series
;
Remez algorithm
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract WithE 2n (|x|) denoting the error of best uniform approximation to |x| by polynomials of degree at most 2n on the interval [−1, +1], the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constantβ for which lim 2nE 2n (|x|)=β.n→∞ Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds forβ: 0.278〈β〈0.286. Now, the average of these bounds is 0.282, which, as Bernstein noted as a “curious coincidence,” is very close to 1/(2√π)=0.2820947917... This observation has over the years become known as the Bernstein Conjecture: Isβ=1/(2√π)? We show here that the Bernstein conjecture isfalse. In addition, we determine rigorous upper and lower bounds forβ, and by means of the Richardson extrapolation procedure, estimateβ to approximately 50 decimal places.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01890040
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