On the best convergence rate of lightning plus polynomial approximations

Abstract:

Building on exponentially clustered poles, Trefethen and his collaborators introduced lightning algorithms for approximating functions of singularities. These schemes are rational approximations with preassigned poles and may achieve root-exponential convergence rates. In particular, based on a specific choice of the parameter of the tapered exponentially clustered poles, the lightning approximation with a low-degree polynomial basis may achieve the optimal convergence rate simply as the best rational approximation explored by Stahl for prototype $x^\alpha$ on $[0,1]$, which was illustrated through delicate numerical experiments and conjectured by Herremans, Huybrechs, and Trefethen [SIAM J. Numer. Anal. 61 (2023), pp. 2580-2600]. By utilizing Poisson’s summation formula and results akin to Paley-Wiener theorem, we rigorously show that all these schemes with a low-degree polynomial basis achieve root-exponential convergence rates with exact orders in approximating $x^\alpha$ for arbitrary clustered parameters theoretically, and provide the best choice of the parameter to achieve the fastest convergence rate for each type of clustered poles, from which the conjecture is confirmed as a special case. Ample numerical evidences demonstrate the optimality and sharpness of the estimates.