Bernstein,s Constant
المؤلف:
Bernstein, S. N.
المصدر:
"Sur la meilleure approximation de |x| par les polynomes de degrés donnés." Acta Math. 37
الجزء والصفحة:
...
19-2-2020
1983
Bernstein's Constant
Let 
be the error of the best uniform approximation to a real function 
on the interval ![[-1,1]](http://mathworld.wolfram.com/images/equations/BernsteinsConstant/Inline3.gif)
by real polynomials of degree at most 
. If
 |
(1)
|
then Bernstein showed that
 |
(2)
|
He conjectured that the lower limit (
) was 
. However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed
 |
(3)
|
For rational approximations 
for 
and 
of degree 
and 
, D. J. Newman (1964) proved
 |
(4)
|
for 
. Gonchar (1967) and Bulanov (1975) improved the lower bound to
 |
(5)
|
Vjacheslavo (1975) proved the existence of positive constants 
and 
such that
 |
(6)
|
(Petrushev 1987, pp. 105-106). Varga et al. (1993) conjectured and Stahl (1993) proved that
 |
(7)
|
REFERENCES:
Bernstein, S. N. "Sur la meilleure approximation de 
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 
." 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 
." 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 
on ![[-1,1]](http://mathworld.wolfram.com/images/equations/BernsteinsConstant/Inline19.gif)
." Russian Acad. Sci. Sb. Math. 76, 461-487, 1993.
Stahl, H. Uniform Rational Approximation of 
. 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 
on ![[-1,+1]](http://mathworld.wolfram.com/images/equations/BernsteinsConstant/Inline22.gif)
. Mat. Sbornik 182, 1523-1541, 1991. English translation in Math. USSR Sbornik 74, 271-290, 1993.
Vjacheslavo, N. S. "On the Uniform Approximation of 
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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