| 释义 |
Bernstein's ConstantN.B. A detailed on-line essay by S. Finchwas the starting point for this entry.
Let  be the error of the best uniform approximation to a Real function  on theInterval  by Real Polynomials of degree at most  . If
 | (1) |
 | (2) |
 ) was  . However, this was disproven by Varga andCarpenter (1987) and Varga (1990), who computed
 | (3) |
 for  and  of degree  and , D. J. Newman (1964) proved
 | (4) |
 . Gonchar (1967) and Bulanov (1975) improved the lower bound to
 | (5) |
 such that
 | (6) |
 | (7) |
References
Bulanov, A. P. ``Asymptotics for the Best Rational Approximation of the Function Sign  .'' Mat. Sbornik 96, 171-178, 1975.Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/brnstn/brnstn.html 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 .'' Russian Acad. Sci. Sb. Math. 76, 461-487, 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  . 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.
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