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

We show that if is the sequence of all zeros of the L-function satisfying then any function from span satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Qn(f,x) of degree at most n such that and for every x∈[-1,1],limn→∞(|f(x)-Qn(f,x)|)/En(f)=0, where En(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, , at the Chebyshev nodes. We also establish a new representation for |L(x,χ)|.