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Bernstein's Constant

Let E_n(f) be the error of the best uniform approximation to a real function f(x) on the interval [-1,1] by real polynomials of degree at most n. If

alpha(x)=|x|,
(1)

then Bernstein showed that

0.267...<lim_(n->infty)2nE_(2n)(alpha)<0.286.
(2)

He conjectured that the lower limit (beta) was beta=1/(2sqrt(pi)). However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed

beta=0.2801694990....
(3)

For rational approximations p(x)/q(x) for p and q of degree m and n, D. J. Newman (1964) proved

1/2e^(-9sqrt(n))<=E_(n,n)(alpha)<=3e^(-sqrt(n))
(4)

for n>=4. Gonchar (1967) and Bulanov (1975) improved the lower bound to

e^(-pisqrt(n+1))<=E_(n,n)(alpha)<=3e^(-sqrt(n)).
(5)

Vjacheslavo (1975) proved the existence of positive constants m and M such that

m<=e^(pisqrt(n))E_(n,n)(alpha)<M
(6)

(Petrushev 1987, pp. 105-106). Varga et al. (1993) conjectured and Stahl (1993) proved that

lim_(n->infty)e^(pisqrt(2n))E_(2n,2n)(alpha)=8.
(7)

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References

Bernstein, S. N. "Sur la meilleure approximation de |x| par les polynomes de degrés donnés." Acta Math. 37, 1-57, 1913.Bulanov, A. P. "Asymptotics for the Best Rational Approximation of the Function Sign x." Mat. Sbornik 96, 171-178, 1975. English translation in Math. USSR Sbornik 5, 275-290, 1968.Finch, S. R. "Bernstein's Constant." §4.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 257-259, 2003.Gonchar, A. A. "Estimates for the Growth of Rational Functions and their Applications." Mat. Sbornik 72, 489-503, 1967.Newman, D. J. "Rational Approximation to |x|." Michigan Math. J. 11, 11-14, 1964.Petrushev, P. P. and Popov, V. A. Rational Approximation of Real Functions. New York: Cambridge University Press, 1987.Stahl, H. "Best Uniform Rational Approximation of |x| on [-1,1]." Russian Acad. Sci. Sb. Math. 76, 461-487, 1993.Stahl, H. Uniform Rational Approximation of |x|. New York: Springer-Verlag, pp. 110-130, 1993.Varga, R. S. Scientific Computations on Mathematical Problems and Conjectures. Philadelphia, PA: SIAM, 1990.Varga, R. S. and Carpenter, A. J. "On a Conjecture of S. Bernstein in Approximation Theory." Math. USSR Sbornik 57, 547-560, 1987.Varga, R. S.; Ruttan, A.; and Carpenter, A. J. "Numerical Results on Best Uniform Rational Approximations to |x| on [-1,+1]. Mat. Sbornik 182, 1523-1541, 1991. English translation in Math. USSR Sbornik 74, 271-290, 1993.Vjacheslavo, N. S. "On the Uniform Approximation of |x| by Rational Functions." Dokl. Akad. Nauk SSSR 220, 512-515, 1975. English translation in Soviet Math. Dokl. 16, 100-104, 1975.

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Bernstein's Constant

Cite this as:

Weisstein, Eric W. "Bernstein's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernsteinsConstant.html

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