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：在这篇论文中，我们提出了拉伽尔谱逼近解析函数的全面收敛分析。通过利用复分析中的积分技巧，我们严格证明了次数为 $n$ 的拉伽尔投影和插值方法在基于原点且顶点在 $z=-\rho^2$ 的抛物线内外存在解析函数的情况下，以根指数速率 $O(\exp(-2\rho\sqrt{n}))$ 收敛，其中 $\rho>0$。我们还讨论了一些重要应用的拓展情况，包括拉伽尔谱微分，高斯 - 拉伽尔积分规则和 Laplace 变换的 Weeks 方法，并得出了一些尖锐的收敛速率估计。我们进行了数值实验来验证理论结果。