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Building on introducing exponentially clustered poles, Trefethen and hiscollaborators introduced lightning algorithms for approximating functions ofsingularities. These schemes may achieve root-exponential convergence rates. Inparticular, based on a specific choice of the parameter of the taperedexponentially clustered poles, the lightning approximation with either alow-degree polynomial basis may achieve the optimal convergence rate simply asthe best rational approximation for prototype $x^\alpha$ on $[0,1]$, which wasillustrated through delicate numerical experiments and conjectured in [SIAM J.Numer. Anal., 61:2580-2600, 2023]. By utilizing Poisson's summation formula andresults akin to Paley-Wiener Theorem, we rigorously show that all these schemeswith a low-degree polynomial basis achieve root-exponential convergence rateswith exact orders in approximating $x^\alpha$ for arbitrary clusteredparameters theoretically, and provide the best choices of the parameter toachieve the fastest convergence rate for each type of clustered poles, fromwhich the conjecture is confirmed as a special case. Ample numerical evidencesdemonstrate the optimality and sharpness of the estimates.

On the best convergence rates of lightning plus polynomial approximations