Hermite polynomials and functions have extensive applications in scientific and engineering problems. While it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, some literature still cast doubts on whether the Hermite spectral method is inferior to other methods. To dispel these doubts, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Moreover, proper scaling can render the Hermite spectral methods comparable to other methods. To illustrate this idea in more detail, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the $L^2$ projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation error. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing pre-asymptotic sub-geometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.

翻译：Hermite多项式与函数在科学与工程问题中具有广泛应用。尽管人们认识到采用尺度化Hermite函数相较于标准函数能显著提升逼近性能，但对尺度因子的理解仍不充分。由于缺乏理论分析，部分文献仍对Hermite谱方法是否劣于其他方法存疑。为消除这些疑虑，本文证明Hermite谱方法的低效性源于目标函数在空间域与频率域衰减速度的不平衡。此外，恰当的尺度化能使Hermite谱方法与其他方法相媲美。为更详细阐述这一观点，我们提出了尺度化Hermite逼近的新型误差分析框架。以$L^2$投影误差为例，该框架表明误差包含三个不同分量：空间截断误差、频率截断误差及Hermite谱逼近误差。通过这一视角，寻找最优尺度因子等价于平衡空间与频率截断误差。作为应用，我们证明对一类函数通过恰当尺度化可恢复几何收敛性。进一步，我们展示对于具有代数衰减的光滑函数，恰当尺度化可使收敛阶数翻倍。该框架能完美解释逼近代数衰减函数时令人困惑的渐近前次几何收敛现象。