Bernstein's constant

# Bernstein's constant

Discussion

Encyclopedia
 BinaryBinary numeral systemThe binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2... 0.01000111101110010011000000110011… DecimalDecimalThe decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations.... 0.280169499… HexadecimalHexadecimalIn mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen... 0.47B930338AAD… Continued fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

Bernstein's constant, usually denoted by the Greek letter β (beta
Beta (letter)
Beta is the second letter of the Greek alphabet. In Ancient Greek, beta represented the voiced bilabial plosive . In Modern Greek, it represents the voiced labiodental fricative ....

), is a mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

named after Sergei Natanovich Bernstein
Sergei Natanovich Bernstein
Sergei Natanovich Bernstein was a Russian and Soviet mathematician known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory.-Partial differential equations:...

and is approximately equal to 0.2801694990.

## 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|, showed that the limit

called Bernstein's constant, exists and is between 0.278 and 0.286. His conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

that the limit is:

was disproven by , who calculated