The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

Abstract

In this paper, we establish new asymptotic relations for the errors of approximation in L p ½À1; 1; 05p41; of jxj l ; l > 0; by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

“…Here we extend the asymptotic formula for s (x) to a complex s with Re s > 0. This asymptotic for a real s > 0 was found in [9]. The proof of a weaker version of (1.1) for s > 0 was outlined in [3, p. 100] (see also [14]).…”

“…Revers [21] established an upper estimate for s C[ −1,1] , where s ∈ (0, 2/3) ∪ {1}. An asymptotic formula for s L p [−1,1] , where s > 0 and 0 < p ∞, was found in [9]. The case p = ∞ was discussed in [3,14].…”

“…The function |x| s , s > 0, has played an important role in approximation and interpolation theory [1][2][3]9,14,21,28,29]. In particular, the asymptotic behavior of the interpolation error s (x) := |x| s − L 2n (f s , x) as n → ∞, where s > 0 and x ∈ [−1, 1], was studied by Bernstein [3], the author [9], and Lubinsky [14]. Revers [21] established an upper estimate for s C[ −1,1] , where s ∈ (0, 2/3) ∪ {1}.…”

On the Bernstein Constants of Polynomial Approximation

“…He proved, using sophisticated methods of potential theory and other complex analytic tools, that Although Λ * α,∞ is not known explicitly, the ideas of Bernstein have been refined, and greatly extended. M. Ganzburg has shown limit relations of this type for large classes of functions, in one and several variables, even when weighted norms are involved [9], [10]. He and others such as Nikolskii and Raitsin have considered not only uniform, but also L p norms.…”

“…He and others such as Nikolskii and Raitsin have considered not only uniform, but also L p norms. It is known [10] that for 1 ≤ p ≤ ∞, there exists…”