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

We show that if {Sk}(k=1)(infinity) is the sequence of all zeros of the L-function L(s, chi) := Sigma(infinity)(k=0)(-1)(k)(2k +1)(-s) satisfying Res(k) epsilon (0, 1), k = 1, 2, then any function from span {vertical bar x vertical bar(sk)}(k=1)(infinity) satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q(n) (f, x) of degree at most it such that parallel to f-Q(n) parallel to C vertical bar-1.1(vertical bar) >><= C(f) E-n (f), n = 1, 2,..., and for every x epsilon vertical bar-1, 1 vertical bar, lim(n ->infinity) (vertical bar.f(x)-Q(n) (f.x)vertical bar)/E(n()f)=0 , where E-n (f) is the error of best polynomial approximation of f in C vertical bar -1, 1 vertical bar. The proof is based on Lagrange polynomial interpolation to vertical bar x vertical bar(x), Res > 0, at the Chebyshev nodes. We also establish a new representation for vertical bar L(x, chi)vertical bar. (C) 2008 Elsevier Inc. All rights reserved.