Bernstein's Constant

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

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

$$\alpha(x)=\vert x\vert,$$ (1)

then Bernstein showed that

$$0.267\ldots < \lim_{n\to\infty} 2nE_{2n}(\alpha)<0.286.$$ (2)

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

$$\beta=0.2801694990\ldots.$$ (3)

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

$${\textstyle{1\over 2}}e^{-9\sqrt{n}}\leq E_{n,n}(\alpha)\leq 3e^{-\sqrt{n}}$$ (4)

for $n\geq 4$. Gonchar (1967) and Bulanov (1975) improved the lower bound to

$$e^{-\pi\sqrt{n+1}}\leq E_{n,n}(\alpha)\leq 3e^{-\sqrt{n}}.$$ (5)

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

$$m\leq e^{\pi\sqrt{n}} E_{n,n}(\alpha)<M$$ (6)

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

$$\lim_{n\to\infty} e^{\pi\sqrt{2n}} E_{2n,2n}(\alpha)=8.$$ (7)

References

Bulanov, A. P. ``Asymptotics for the Best Rational Approximation of the Function Sign $x$.'' 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 $\vert x\vert$.'' 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 $\vert x\vert$ on $[-1,1]$.'' 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 $\vert x\vert$ on $[-1,+1]$. Math. USSR Sbornik 74, 271-290, 1993.

Vjacheslavo, N. S. ``On the Uniform Approximation of $\vert x\vert$ by Rational Functions.'' Dokl. Akad. Nauk SSSR 220, 512-515, 1975.