In this paper, we present a comprehensive convergence analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove rigorously 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 $z=-\rho^2$. The extension to several important applications are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules 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：本文对解析函数的拉盖尔谱逼近进行了全面的收敛性分析。通过利用复分析中的围道积分技术，我们严格证明了：当所考虑的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，$n$ 次拉盖尔投影与插值方法以根指数速率 $O(\exp(-2\rho\sqrt{n}))$（$\rho>0$）收敛。本文进一步讨论了该方法在若干重要应用中的推广，包括拉盖尔谱微分、高斯-拉盖尔求积规则以及用于拉普拉斯变换反演的威克斯方法，并导出了若干精确的收敛速率估计。数值实验验证了理论结果。