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Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

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Abstract

When a function is expanded as \(f(x) \approx \sum \limits _{n=0}^{N-1}\, a_{n}\phi_{n}(x)\)for some set of basis functions \({\phi_{j}(x)}\) , its spectral coefficients a n generally have an asymptotic approximation, as n→∞, in the form of an inverse power series plus terms that decrease exponentially with n. If f(x) is analytic on the expansion interval, then all the coefficients of the inverse power series are zero and the problem becomes one of “beyond-all-orders” or “exponential” asymptotics. The method of steepest descent for integrals and other complex-path integration techniques can successfully connect the exponentially small behavior of the spectral coefficients to the singularities of f(x) off the expansion interval. Many examples are given in both one and two dimensions.

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  1. Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI, 48109-2143, USA

    John P. Boyd

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Boyd, J.P. Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms. J Eng Math 63, 355–399 (2009). https://doi.org/10.1007/s10665-008-9241-3

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