How well does the Hermite–Padé approximation smooth the Gibbs phenomenon?

Abstract:

In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function $f$, defined on $[-\pi ,\pi ]$, discontinuous at 0,

Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants of the system of functions $(1,g_{1} (z),g_{2} (z))$, where $g_{1} (z)=\log (1-z)$ and $g_{2} (z)$ is analytic, such that $\operatorname {Re}(g_{2} (e^{it}))=f (t)$. Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Padé approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Padé approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Padé approximants are more efficient than the more elementary Padé approximants, particularly around the discontinuity of the goal function $f$.