Asymptotic approximations elucidating the Gibbs phenomenon and Fejér averaging

Abstract

This paper deals with two trigonometric sums that are pervasive in the literature dealing with the Gibbs phenomenon. In particular, the two sums often serve as test cases for methods – such as the method of Fejér averaging – that aim to overcome the Gibbs phenomenon. Each of the two is the partial sum of a convergent infinite series with a discontinuous limit function. Our starting points are some recently-published results, both exact and asymptotic, for the two sums. For the case of a large number of terms, we proceed from those results to develop simple and revealing asymptotic formulas for the two sums and, also, for their Fejér averages. These formulas break down as we approach the point of discontinuity, so we further develop similar formulas that are appropriate near the discontinuity point. We repeat all these tasks for a third sum whose corresponding infinite series exhibits a logarithmic singularity. Our asymptotic results, which view the three sums from a new perspective, illuminate many aspects of the Gibbs phenomenon and the Fejér averaging method. As an illustrative example of the applicability of our formulas, we exploit their properties to develop a convergence acceleration method. We then use this method to accelerate some (more complicated) sums that exhibit logarithmic singularities, including a sum that arises in several physics applications.