On the Bernstein Constants of Polynomial Approximation

Abstract

Let α > 0 not be an integer. In papers published in 1913 and 1938, S. N. Bernstein established the limit.denotes the error in best uniform approximation of |x| α on [−1, 1] by polynomials of degree ≤ n. Bernstein proved that Λ * ∞,α is itself the error in best uniform approximation of |x| α by entire functions of exponential type at most 1, on the whole real line. We prove that the best approximating entire function is unique, and satisfies an alternation property. We show that the scaled polynomials of best approximation converge to this unique entire function. We derive a representation for Λ

Extremal Polynomials and Entire Functions of Exponential Type

“…In his 1938 paper, Bernstein made essential use of the homogeneity property of |x| α , namely that for c > 0 one has |cx| α = c α |x| α . Using this property, one gets for a, b > 0 and all 1 ≤ p ≤ ∞ the relation (see [10], Lemma 8.2)…”

“…Especially we are interested into the locations of its corresponding interpolation points. Recall, that from ( [10], [11]) it follows, that uniformly on compact subsets of C we have There is also a representation for H * α as an interpolation series with (unknown) interpolation points 0 < x * 1 < x * 2 < x * 3 < · · · . However, it is known ( [11], Theorem 1.1) that…”

Best Rational Approximation of Functions with Logarithmic Singularities

“…In contrast to this, it is not known whether the Bernstein constant b(α) in (1.1) can be expressed in terms of standard transcendentals; see, e.g. [12] for more on this issue. 2.…”

L1-approximation to Laplace transforms of signed measures

“…An extension of the Krein-Nagy method has been given recently in Ganzburg [6]. For approximations in different norms, we refer to Lubinsky [13] and Ganzburg and Lubinsky [7]. For the analogous problem on the torus, see Ganelius [5].…”