Optimal convergence analysis of Laguerre spectral approximations for analytic functions
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In this paper, we present a comprehensive convergence analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove rigorously that Laguerre projection and interpolation methods of degree n converge at the root-exponential rate O(exp(-2ρ√(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. The extension to several important applications are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules 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.
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