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Trefethen and his collaborators in \cite{Herremans2023} built further on alightning approximation with a low-degree polynomial basis based on introducinga new choice of the tapered exponentially clustering poles. The new schemeachieves the optimal convergence rate for approximation of $x^\alpha$ on$[0,1]$ as by the best rational approximation shown by Stahl in 2003,illustrated through ample delicate numerical experiments. This paper, with theaid of Poisson summation formula, Cauchy's integral theorem and the decaybehaviors of Fourier transforms, rigorously shows that the new scheme,utilizing tapered exponentially clustering poles, achieves a root-exponentialconvergence rate in approximating $x^\alpha$, from which Conjecture 3.1 in\cite{Herremans2023} is confirmed, and the choice of the parameter$\sigma=\frac{2\pi}{\sqrt{\alpha}}$ achieves the fastest convergence rate amongall $\sigma>0$.

On the best convergence rate of lightning plus polynomial approximation for $x^{\alpha}$