The Rate of Convergence of Fourier Coefficients for Entire Functions of Infinite Order with Application to the Weideman-Cloot Sinh-Mapping for Pseudospectral Computations on an Infinite Interval

Abstract

We analytically compute the asymptotic Fourier coefficients for several classes of functions to answer two questions. The numerical question is to explain the success of the Weideman-Cloot algorithm for solving differential equations on an infinite interval. Their method combines Fourier expansion with a change-of-coordinate using the hyperbolic sine function. The sinh-mapping transforms a simple function like exp(-z2) into an entire function of infinite order. This raises the second, analytical question: What is the Fourier rate of convergence for entire functions of an infinite order? The answer is: Sometimes even slower than a geometric series. In this case, the Fourier series converge only on the real axis even when the function u (z) being expanded is free of singularities except at infinity. Earlier analysis ignored stationary point contributions to the asymptotic Fourier coefficients when u(z) had singularities off the real z-axis, but we show that sometimes these stationary point terms are more important than residues at the poles of u(z).