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.

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