Convergence analysis of Laguerre approximations for analytic functions

TL;DR

This paper provides the first rigorous proof of root-exponential convergence rates for Laguerre spectral approximations of analytic functions, using complex analysis techniques, with applications to spectral differentiation, quadrature, and Laplace transform inversion.

Contribution

It introduces a comprehensive convergence analysis for Laguerre spectral methods, establishing root-exponential rates for analytic functions and exploring various practical applications.

Findings

Laguerre spectral approximations converge at root-exponential rate for analytic functions.

The analysis applies contour integral techniques from complex analysis.

Numerical experiments confirm the theoretical convergence rates.

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 $O(\exp(-2\rho\sqrt{n}))$ with $\rho>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-\rho^2$. 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.