In this paper, we present a rigorous analysis of root-exponential convergence of Hermite approximations, including projection and interpolation methods, for functions that are analytic in an infinite strip containing the real axis and satisfy certain restrictions on the asymptotic behavior at infinity within this strip. Asymptotically sharp error bounds in the weighted and maximum norms are derived. The key ingredients of our analysis are some remarkable contour integral representations for the Hermite coefficients and the remainder of Hermite spectral interpolations. Further extensions to Gauss--Hermite quadrature, Hermite spectral differentiations, generalized Hermite spectral approximations and the scaling factor of Hermite approximation are also discussed. Numerical experiments confirm our theoretical results.

翻译：本文对Hermite逼近（包括投影法和插值法）在解析函数中的根指数收敛性进行了严格分析，这些函数在包含实轴的无限带形区域内解析，并满足该区域内无穷远处渐近行为的特定限制。我们导出了加权范数和最大范数下的渐近精确误差界。分析的关键工具是Hermite系数及Hermite谱插值余项的几个显著围道积分表示。此外，还讨论了向Gauss-Hermite求积、Hermite谱微分、广义Hermite谱逼近及Hermite逼近尺度因子的扩展应用。数值实验验证了我们的理论结果。