Distribution of interpolation points of bestL 2-approximants (nth partial sums of Fourier series)

Abstract

For a continuous 2π-periodic real-valued functionf, we investigate the asymptotic behavior of the zeros of the errorf(θ)−sn(θ), wheresn(θ) is thenth Fourier section. We prove that there is a subsequence {nk} for which such zeros (interpolation points) are uniformly distributed on [−π, π]. This extends previous results of Saff and Shekhtman. Moreover, results dealing with the maximal distance between consecutive zeros off−snk are obtained. The technique of proof involves coefficient estimates for lacunary trigonometric polynomials in terms of itsLq-norm on a subinterval.