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

Abstract

We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s, ) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2, . . . , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f )E n (f ), n=1, 2, . . . , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f )= 0, where E n (f ) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x, )|.

Extremal Polynomials and Entire Functions of Exponential Type

“…Firstly, we mention that in [2] Bernstein himself established a slightly weaker solution compared to formula (1.5). Secondly, an extension of limit relation (1.5) to complex values for α was obtained recently in [6].…”

Polynomial interpolation and asymptotic representations for zeta functions

“…Results of Chapter 2 generalize and extend integral interpolation formulae by Hermite [26], Bernstein [5, pp. 92, 98], Lubinsky [38], and the author [19,20,21,23].…”

“…92, 98] was the first author who extended this formula for Lagrange interpolation to the nonanalytic function f (y) = (1 − y) s , s > 0, on [−1 , 1]. Various versions of Bernstein's result were discussed by the author [19,20,21,23]. Lubinsky [38, [9] eq.…”

“…General nodes. Historically, the function |y| s plays an important role in Lagrange interpolation and polynomial approximation; see for example [5,6,19,20,38,21], where further references are given.…”

“…Note that an identity like (2.3.17) for s > 0 and Lagrange interpolation was established by Bernstein [5, p. 98] who used (2.1.2) and the Cauchy theorem to prove his result. Various versions of Bernstein's identity were found by the author [19,20,21,23]. In particular, the case of 0 < Re s < 2N and the Chebyshev nodes was discussed in [21].…”