Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree $n$ converge at the root-exponential rate $O(\exp(-2\rho\sqrt{n}))$ with $\rho>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-\rho^2$. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.

翻译：拉盖尔谱逼近在无界域问题的高效算法发展中具有重要作用。本文对解析函数的拉盖尔谱逼近进行了全面的收敛速率分析。通过利用复分析中的围道积分技术，我们证明了当目标函数在以原点为焦点、顶点位于$z=-\rho^2$的抛物线内部及边界上解析时，次数为$n$的拉盖尔投影与插值方法以根指数速率$O(\exp(-2\rho\sqrt{n}))$（其中$\rho>0$）收敛。据我们所知，这是拉盖尔逼近用于解析函数时根指数收敛性的第一个严格证明。本文还讨论了该分析的若干重要应用，包括拉盖尔谱微分、高斯-拉盖尔求积法则、拉普拉斯变换反演中的缩放因子与Weeks方法，并推导出若干精确的收敛速率估计。数值实验验证了理论结果。