Bernstein's constant $\beta$
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Number
$\beta$
0.28016949902386913303643649123067200004248213981236
Definition
Bernstein's constant is $\beta=\lim_{n\to\infty}2nE_{2n}$, where $E_m=\inf_{p\in\mathbb R[x],\,\deg p\leq m}\max_{-1\leq x\leq1}\bigl||x|-p(x)\bigr|$ is the error of best uniform approximation of $|x|$ on $[-1,1]$ by real polynomials of degree at most $m$ [1] [3].
Formulas
(1)
$E_{2n+1}=E_{2n}$ for $n\geq0$, and $\beta=\lim_{m\to\infty}mE_m$ [1].
(2)
$0.2801685<\beta<0.2801734$, by outward rounding of the rigorous bounds in equation (1.16) of [1].
Comments
(3)
Symmetrising a polynomial cannot increase its uniform error against the even function $|x|$, so an optimal polynomial of degree at most $2n+1$ can be chosen even and has degree at most $2n$. The substitution $t=x^2$ also identifies $E_{2n}$ with the error of best degree-$n$ polynomial approximation of $\sqrt t$ on $[0,1]$ [1] [2].
(4)
Bernstein conjectured $\beta=1/(2\sqrt\pi)$. Varga and Carpenter disproved this equality by bounding $\beta$ as in (2) [1].
References
[1]
R. S. Varga and A. J. Carpenter, On the Bernstein conjecture in approximation theory, Constructive Approximation 1 (1985), 333–348. (doi)
[2]
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 4.4, pp. 257–259.
Links
Similar tables
Lebesgue constants of the Fourier partial sums —   norms of Fourier projection operators in uniform approximation of periodic functions
Bernstein basis polynomials —   a constructive family for polynomial approximation, also named after Bernstein
Data properties
Entries are of type: real number
Table is complete: yes (it holds the single constant in the polynomial-degree normalisation on $[-1,1]$)
How they were obtained:

The 50 decimal places are transcribed from equations (1.18) and (5.3) of Varga and Carpenter [1]; a scan is available as [5]. They computed the minimax errors $E_{2n}$ for $1\leq n\leq52$ by the Remez algorithm at 100 decimal working digits and applied Richardson extrapolation in $1/n^2$ to $2nE_{2n}$. Their finite-degree errors were estimated accurate to at least 95 decimal places. Of 210 entries in the last 20 extrapolation columns, 182 agreed with all 50 reported digits. These are heuristic digits of the limit, not a certified 50-digit enclosure. No additional digits are asserted.

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The transcription matches all 50 digits of [4], which records the same source computation. A separately implemented calculation at 160 decimal working digits recomputes all 52 minimax errors and the Richardson tableau. Its control is the exact degree-two minimax polynomial $x^2+1/8$, whose uniform error is $1/8$. For each degree it checks alternating residuals, locates all positive critical points, and requires the spread of their error magnitudes to be less than $10^{-90}$. Five finite-degree errors are also compared with the 20-place values in Table 2.1 of the paper.

The independent extrapolation differs from the transcribed decimal by about $1.3932\times10^{-51}$, within its one-unit last-place interval. The attached checks compare this independent result and the OEIS digits with the value read back from the database. Agreement of extrapolations does not supply a remainder bound, so the rigour remains heuristic.