Hermite polynomials and functions are widely used for scientific and engineering problems. Although it is known that using the scaled Hermite function instead of the standard one can significantly enhance approximation performance, understanding of the scaling factor is inadequate. To this end, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the $L^2$ projection error as an example, our results illustrate that when using truncated $N$ terms of scaled Hermite functions to approximate a function, there are three different components of error: spatial truncation error; frequency truncation error; and Hermite spectral approximation error. Through our insight, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation error. As an example, 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 puzzling pre-asymptotic sub-geometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.

翻译：埃尔米特多项式与函数在科学与工程问题中应用广泛。尽管已知使用尺度化埃尔米特函数替代标准函数能显著提升逼近性能，但关于尺度因子的理解尚不充分。为此，我们提出了一种针对尺度化埃尔米特逼近的新型误差分析框架。以$L^2$投影误差为例，我们的结果表明：当使用截断的$N$项尺度化埃尔米特函数逼近某函数时，误差包含三个不同分量：空间截断误差、频率截断误差以及埃尔米特谱逼近误差。通过我们的分析发现，寻找最优尺度因子等价于平衡空间截断误差与频率截断误差。作为示例，我们证明对于某类函数，通过适当尺度化可恢复几何收敛性。此外，我们证明对于具有代数衰减的光滑函数，适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。