Let En(ƒ) be the error of the best uniform approximation to a real functionƒ(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
^Bernstein, S.N. (1914). "Sur la meilleure approximation de x par des polynomes de degrés donnés". 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. Bibcode:1987SbMat..57..547V. doi:10.1070/SM1987v057n02ABEH003086. MR 0842399.
bernstein, constant, usually, denoted, greek, letter, beta, mathematical, constant, named, after, sergei, natanovich, bernstein, equal, 2801694990, binary, 01000111101110010011000000110011, decimal, 280169499, hexadecimal, 47b930338aad, continued, fraction, di. Bernstein s constant usually denoted by the Greek letter b beta is a mathematical constant named after Sergei Natanovich Bernstein and is equal to 0 2801694990 1 Binary 0 01000111101110010011000000110011 Decimal 0 280169499 Hexadecimal 0 47B930338AAD Continued fraction 1 3 1 1 1 1 1 3 1 9 displaystyle cfrac 1 3 cfrac 1 1 cfrac 1 1 cfrac 1 3 cfrac 1 9 ddots Definition EditLet 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 b lim n 2 n E 2 n f displaystyle beta lim n to infty 2nE 2n f called Bernstein s constant exists and is between 0 278 and 0 286 His conjecture that the limit is 1 2 p 0 28209 displaystyle frac 1 2 sqrt pi 0 28209 dots was disproven by Varga and Carpenter 3 who calculated b 0 280169499023 displaystyle beta 0 280169499023 dots References Edit sequence A073001 in the OEIS Bernstein S N 1914 Sur la meilleure approximation de x par des polynomes de degres donnes 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 Bibcode 1987SbMat 57 547V doi 10 1070 SM1987v057n02ABEH003086 MR 0842399 Further reading EditWeisstein Eric W Bernstein s Constant MathWorld Retrieved from https en wikipedia org w index php title Bernstein 27s constant amp oldid 1117461365, wikipedia, wiki, book, books, library,
