Convergence analysis of Laguerre approximations for analytic functions

TL;DR

This work establishes a rigorous framework for the convergence of Laguerre approximations of analytic functions on unbounded domains by exploiting contour integrals over a parabolic region $P_{\rho}$. The authors prove root-exponential decay, $O(\exp(-2\rho\sqrt{n}))$, for Laguerre coefficients and the associated projection/interpolation errors, for both generalized Laguerre polynomials and generalized Laguerre functions, and extend the analysis to practical tools such as Gauss-Laguerre quadrature and the Weeks method for Laplace inversion. They also derive sharp error estimates for Laguerre spectral differentiations and discuss scaling strategies to accelerate convergence. Numerical experiments corroborate the theoretical rates and demonstrate the practical impact on unbounded-domain problems, with potential extensions to Hermite approximations planned for future work.

Abstract

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree $n$ converge at the root-exponential rate $O(\exp(-2ρ\sqrt{n}))$ with $ρ>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-ρ^2$. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.

Figures (8)

• Figure 1: Left: The parabola $P_{\rho}$ and the region $D_{\rho}$. The two points on the horizontal axis (from left to right) are the vertex and focus of $P_{\rho}$, respectively. Right: The dashed lines indicates the five contours used in the proof of Theorem \ref{['thm:LagParab']}.
• Figure 2: The magnitudes of the Laguerre coefficients of $f_1(x)=1/(x+1)$ (squares), $f_2(x)=e^{-x}/(4x+9)$ (circles) and $f_3(x)=\mathrm{sech}(\pi{x}/16)$ (dots). Here $\alpha=0$ (left) and $\alpha=3/2$ (right) and the dashed lines show the corresponding predicted rates $O(n^{-\alpha/2-1/4}e^{-2\rho\sqrt{n}})$.
• Figure 3: The maximum errors of $\Pi_n^{\mathrm{F}}f$ and the predicted convergence rates (dashed) for $f(x)=e^{-x/2}/(1+x)$ (left) and $f(x)=e^{-2x/3}/(x^2+4)$ (right).
• Figure 4: The errors of the Laguerre interpolants (left) and Laguerre-Radau interpolants (right) for $f(x)=e^{-x}/(x+1)$ (boxes), $f(x)=1/(x^2+9)$ (dots) and $f(x)=\mathrm{sech}(\pi{x}/16)$ (circles). The dashed lines denote the predicted convergence rates.
• Figure 5: The magnitude of the Laguerre coefficients of $f(x)$. The left panel shows $f(x)=\cos(x)$ (circles), $f(x)=e^{-x}\sin(x)$ (boxes) and $f(x)=e^{-x}$ (dots) and the right panel shows $f(x)=e^{-x^2}$ (circles), $f(x)=e^{-x^2}\sin(x)$ (boxes) and $f(x)=e^{-x^2}\cos(x)$ (dots). The dashed line in the right panel is $e^{-\kappa{n}^{2/3}}$ with $\kappa=3/5$.

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Theorem 3.1
• proof
• Remark 3.2
• Remark 3.3
• Theorem 3.4
• proof
• Remark 3.5
• Remark 3.6