Scaling Optimized Spectral Approximations on Unbounded Domains: The Generalized Hermite and Laguerre Methods

TL;DR

This work develops a bandwidth-driven, Nyquist-Shannon–style error analysis framework for scaled generalized Laguerre and Hermite methods on unbounded domains. By linking the sampling nodes to spatial and frequency bandwidths, the authors derive an explicit three-term error decomposition and a systematic rule for choosing the optimal scaling $\beta$, enabling root-exponential and geometric convergence for broad function classes. The framework also provides a detailed, cross-method comparison between Hermite and Laguerre bases, showing that two concatenated Laguerre bases can rival or outperform a single Hermite basis in many settings, and clarifies why pre-asymptotic behavior may appear sub-geometric. These results lead to practical guidance for stable, accurate spectral approximations and quadrature on unbounded domains, with implications for nonlinear PDEs, optimal control, and quantum/kinetic models.

Abstract

We propose a novel error analysis framework for scaled generalized Laguerre and generalized Hermite approximations.This framework can be regarded as an analogue of the Nyquist-Shannon sampling theorem: It characterizes the spatial and frequency bandwidths that can be effectively captured by Laguerre or Hermite sampling points. Provided a function satisfies the corresponding bandwidth constraints, it can be accurately approximated within this framework. The proposed framework is notably more powerful than classical theory -- it not only provides systematic guidance for choosing the optimal scaling factor, but also predicts root-exponential and other intricate convergence behaviors that classical approaches fail to capture. Leveraging this framework, we conducted a detailed comparative study of Hermite and Laguerre approximations. We find that functions with similar decay and oscillation characteristics may nonetheless display markedly different convergence rates. Furthermore, approximations based on two concatenated sets of Laguerre functions may offer significant advantages over those using a single set of Hermite functions.

Figures (6)

• Figure 1: The $L^2$ error $\|u - u_{N,\beta}\|$ of the Laguerre--Galerkin method solving \ref{['eq:0624-18']} with $u = e^{-x^2} - e^{-\frac{x^2}{2}}, \beta=1$.
• Figure 2: The $L^2$ error $\|u - u_{N,\beta}\|$ of the Laguerre--Galerkin method solving \ref{['eq:0624-18']} with $u = e^{-x^2} - e^{-\frac{x^2}{2}}, \beta=N^{1/2}$.
• Figure 3: The $L^2$ error $\|u - u_{N,\beta}\|$ of the Laguerre--Galerkin method solving problem \ref{['eq:0624-18']} with $u = x/(1+x)^2, \beta=10, \beta = 100/N$.
• Figure 4: The $L^2$ error $\|u - u_{N,\beta}\|$ for problem \ref{['eq:0624-18']} with $\gamma = 1$ and true solution $u = x/(1+x^2)^h$. Here $\beta = 1$.
• Figure 5: The $L^2$ error $\|u - u_{N,\beta}\|$ for problem \ref{['eq:0624-18']} with $u = \sin 2x \cdot (1+x)^{-\frac{7}{2}}$ with different $\beta$.

Theorems & Definitions (31)

• Lemma 2.1
• Lemma 2.2
• Lemma 4.1
• Theorem 4.1
• Remark 1
• proof
• Corollary 4.1
• Theorem 5.1
• Remark 2
• Remark 3