Bernstein's constant
| Binary | 0.01000111101110010011000000110011… |
| Decimal | 0=9 |
| Hexadecimal | 0.47B930338AAD… |
| Continued fraction |
Bernstein's constant, usually denoted by the Greek letter β (beta), is a mathematical constant named after Sergei Natanovich Bernstein and is equal to 0.2801694990... .[1]
Definition
Let En(ƒ) be the error of the best uniform approximation to a real function ƒ(x) on the interval [−1, 1] by real polynomials of no more than degree n. In the case of ƒ(x) = |x|, Bernstein[2] showed that the limit
called Bernstein's constant, exists and is between 0.278 and 0.286. His conjecture that the limit is:
was disproven by Varga and Carpenter,[3] who calculated
References
- ^ (sequence A073001 in the OEIS)
- ^ Bernstein, S.N. (1914). "Sur la meilleure approximation de x par des polynomes de degrés donnés" (PDF). Acta Math. 37: 1–57. doi:10.1007/BF02401828.
- ^ Varga, Richard S.; Carpenter, Amos J. (1987). "A conjecture of S. Bernstein in approximation theory". Math. USSR Sbornik. 57 (2): 547–560. doi:10.1070/SM1987v057n02ABEH003086. MR 0842399.