Scaling Optimized Hermite Approximation Methods

Abstract: 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.