On the bernstein conjecture in approximation theory

Abstract

With Ea,,(lx I) denoting the error of best uniform approximation to Ixl by polynomials of degree at most 2n on the interval [-1, + 1J, the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant fl for which lim 2nE2,,(rxJ) =: ft.Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds for fl: 0.278 < fl < 0.286. Now, the average of these bounds is 0.282, which, as Bernstein noted as a "curious coincidence," is very close to 1/(2~/~ -) = 0.2820947917 .... This observation has over the years become known as the Bemstein Conjecture: Is fl = 1/(2r We show here that the Bemstein conjecture is false. In addition, we determine rigorous upper and lower bounds for fl, and by means of the Richardson extrapolation procedure, estimate /~ to approximately 50 decimal places.

Best uniform rational approximation of 𝑥^{𝛼} on [0,1]

“…In [Bel,p. 56] Bernstein raised the question whether ß can be expressed by known transcendental or whether it defines a new one, and based on numerical upper and lower bounds for ß , which he calculated up to a precision of ±0.005, he made the tentative conjecture ß = \/(2^/n), which, however, has been disproved in [VC1] by extensive and nontrivial high precision calculations.…”

“…It may be surprising that in case of rational approximation, which is in many respects more complex than the polynomial case, limit (2.5) has a rational value, while in the polynomial case Bernstein's question in [Bel] about the character of the number ß = ß(\) is still open and the numerical results in [VC1] show that ß cannot be a rational number with a moderately small denominator. [GoLa; StTo, § §6.1, 6.2]).…”

On Interpolation Approximation: Convergence Rates for Polynomial Interpolation for Functions of Limited Regularity

“…(see Bernstein [3] and Varga and Capenter [63]). Thus, the error estimate (1.5) by using the Lebesgue constant may be overestimated for some special points of sets for functions of limited regularities.…”

On the Bernstein Constants of Polynomial Approximation

“…Bernstein did not determine the value of Λ * α,∞ , but speculated that Some 70 years later, this was disproved by Varga and Carpenter [34], [33] using high precision scientific computation. They showed that Λ * ∞,1 = 0.28016 94990 .…”