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

Technical Summary: Scaling Optimized Spectral Approximations on Unbounded Domains

Problem Statement
Spectral methods based on Laguerre and Hermite polynomials are standard tools for solving partial differential equations (PDEs) on unbounded domains (the half-line and the whole real line, respectively). However, their application faces three primary challenges:

1. Numerical Instability: Computing high-degree polynomials in floating-point arithmetic often leads to overflow and round-off errors.
2. Limited Resolution: Direct application yields accurate approximations only over small intervals because classical error estimates are derived in weighted Sobolev spaces with rapidly decaying weights.
3. Suboptimal Resolution: Standard Laguerre and Hermite polynomials exhibit inferior resolution properties compared to orthogonal polynomials on bounded domains.

While scaling factors have been introduced to mitigate these issues, a systematic framework for selecting the optimal scaling factor and explicitly characterizing how approximation error depends on this factor has been lacking. Classical theories, formulated in finite-smoothness function spaces, fail to predict complex convergence behaviors, such as root-exponential convergence for analytic functions, and cannot adequately explain the efficiency of scaled methods.

Methodology
The authors propose a novel error analysis framework for scaled generalized Laguerre and Hermite approximations. This framework is constructed as an analogue to the Nyquist-Shannon sampling theorem.

• Bandwidth Characterization: The method precisely characterizes the spatial (physical) bandwidth and frequency bandwidth that can be effectively captured by the sampling points of Laguerre or Hermite expansions.
  • Spatial Bandwidth ($M$): Defined as $O(\sqrt{N}/\beta)$, representing the interval where the function must decay sufficiently.
  • Frequency Bandwidth ($B$): Defined as $O(\beta\sqrt{N})$, representing the frequency range where the Fourier transform must decay.
• Error Decomposition: The total approximation error is decomposed into three distinct components:
  1. Spatial Truncation Error: Caused by insufficient decay of the target function outside the spatial bandwidth.
  2. Frequency Truncation Error: Caused by insufficient decay of the Fourier transform outside the frequency bandwidth.
  3. Spectral Error: An exponentially decaying error inherent to the approximation within the truncated bandwidth.
• Theoretical Equivalence: The authors establish a mathematical equivalence between generalized Laguerre approximations on the half-line and generalized Hermite approximations on the whole line via variable transformation ($u(x) = v(x^2)$). This allows the derivation of Laguerre error bounds directly from Hermite analysis.
• Extension: The core projection error estimates are extended to cover derivatives, interpolation errors, quadrature errors, and model PDEs (specifically a second-order elliptic problem on the half-line).

Key Contributions and Results

1. Systematic Scaling Guidance: The framework provides a rigorous method for choosing the optimal scaling factor $\beta$. By balancing the spatial and frequency truncation errors, the convergence rate can be significantly improved.

   • For functions exhibiting subgeometric convergence under standard scaling, optimal scaling can accelerate convergence to geometric rates.
   • For functions with algebraic convergence, optimal scaling can double the convergence order.

2. Prediction of Complex Convergence Behaviors: Unlike classical theory, this framework successfully predicts:

   • Root-exponential convergence for analytic functions, attributing it directly to the exponential decay rate of the function's Fourier transform.
   • Pre-asymptotic behaviors: It explains why functions with algebraic decay in the spatial domain may exhibit sub-geometric convergence (e.g., $e^{-c\sqrt{N}}$) in the pre-asymptotic regime. This occurs when the frequency truncation error dominates the spatial truncation error before the asymptotic regime is reached.

3. Comparative Analysis of Hermite vs. Laguerre:

   • The study reveals that functions with similar decay and oscillation characteristics can display markedly different convergence rates depending on whether they are approximated using Hermite or Laguerre bases.
   • Dual Laguerre vs. Single Hermite: The authors compare a single set of Hermite functions against two concatenated sets of Laguerre functions (one for $x \ge 0$ and one for $x < 0$). For a class of analytic functions (specifically $u(x) = e^{-x^2}g(x)$ where $g$ is analytic), the dual Laguerre approach is shown to perform at least as well as, and often better than, the single Hermite set.
   • Specifically, for $u(x) = e^{-x^2}/(1+16x^2)$, the optimal Hermite quadrature achieves a convergence rate of $e^{-cN^{2/3}}$, while the optimally scaled dual Laguerre approach achieves $e^{-cN^{4/5}}$.

4. Quadrature Efficiency: The paper demonstrates that the perceived inefficiency of Hermite/Laguerre Gaussian quadrature compared to bounded-domain quadratures is largely due to a lack of optimal scaling. With the correct scaling factor (e.g., $\beta \sim N^{1/6}$), Hermite quadrature can achieve accuracy comparable to bounded-domain rules (e.g., $e^{-cN^{2/3}}$ for specific oscillatory integrals).

Significance
The paper claims that this framework offers a more powerful and detailed characterization of spectral approximations on unbounded domains than classical theories. Its significance lies in:

• Unifying Theory: It unifies the understanding of scaling, resolution, and convergence by linking them to bandwidth constraints analogous to signal processing.
• Practical Guidance: It moves beyond theoretical existence to provide actionable guidelines for selecting scaling factors to maximize accuracy.
• Explaining Anomalies: It resolves long-standing puzzles in the literature, such as the "puzzling" root-exponential convergence of algebraically decaying functions in pre-asymptotic regimes and the superior performance of dual Laguerre sets over single Hermite sets for certain analytic functions.
• Generalizability: The framework is shown to be applicable to derivatives, interpolation, quadrature, and model PDEs, suggesting a broad utility for numerical analysis on unbounded domains.

The authors conclude that while the dual Laguerre approach shows superior performance in specific tested cases, this may reflect a general principle worthy of further investigation, though they do not claim it as a universal theorem for all functions.