## Bernstein's Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let be the error of the best uniform approximation to a Real function on the Interval 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

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.