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

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.

💡 Research Summary

The paper introduces a comprehensive error‑analysis framework for spectral approximations on unbounded domains using scaled generalized Laguerre functions (GLFs) and scaled generalized Hermite functions (GHFs). The authors view the framework as an analogue of the Nyquist‑Shannon sampling theorem: the set of N + 1 scaled basis functions defines a spatial bandwidth of order √N / β and a frequency bandwidth of order β√N, where β is the scaling factor. A target function that is sufficiently localized both in physical space (decays fast outside √N / β) and in Fourier space (its spectrum decays fast beyond β√N) can be approximated with an error that decomposes into three contributions: (i) spatial truncation error caused by insufficient decay outside the spatial bandwidth, (ii) frequency truncation error caused by high‑frequency components beyond the frequency bandwidth, and (iii) a spectral error that decays exponentially with β²N. By quantifying each term, the framework provides a principled way to choose the optimal β for a given function class.

The core theoretical result (Theorem 4.1) gives an explicit L²‑error bound for the projection onto the first N + 1 scaled GHFs. The bound shows that, when β is tuned to match the function’s spatial and spectral extents, the overall error can exhibit root‑exponential convergence (∼e^{‑cβ²N}) for analytic functions, a doubling of algebraic convergence order for algebraically decaying functions, and more intricate rates for intermediate cases. The analysis relies on precise asymptotics of Hermite polynomial zeros, which determine the sampling points, and on Fourier transform identities that relate the decay of the spectrum to the frequency bandwidth.

A parallel treatment for Laguerre approximations follows from a transformation that maps scaled Laguerre bases to scaled Hermite bases. The authors emphasize that the two families differ fundamentally in how their zeros populate the domain: Hermite zeros are essentially symmetric and densely packed in an interval of width O(√N / β), whereas Laguerre zeros lie on the half‑line and are non‑uniform. Consequently, the spatial bandwidth for Laguerre approximations is also √N / β, but the effective coverage of the whole real line can be enhanced by using two concatenated Laguerre sets (one on each half‑line). Numerical experiments demonstrate that this “dual‑Laguerre” approach often outperforms a single Hermite set, achieving up to 30 % lower L² error for the same N.

The paper also extends the theory to derivatives, interpolation, and Gaussian quadrature. Scaling improves the accuracy of Laguerre‑Gauss and Hermite‑Gauss quadratures dramatically, turning what is otherwise a sub‑optimal rule into one with exponential convergence. Applications to model PDEs (e.g., nonlinear equations on semi‑infinite intervals, Schrödinger equations on the whole line) confirm that optimal scaling accelerates convergence from sub‑geometric to geometric or even root‑exponential rates.

In summary, the authors provide a unified, bandwidth‑based perspective on spectral approximation in unbounded domains. Their framework not only explains previously puzzling convergence phenomena (such as root‑exponential rates for analytic functions and unexpected algebraic acceleration) but also supplies a practical recipe for selecting scaling parameters and for choosing between Hermite and Laguerre bases. The work opens avenues for further extensions to multi‑dimensional unbounded domains, time‑dependent scaling, and more exotic orthogonal systems.