Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

AbstractIn constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval $$x \in [-\chi, \chi]$$, to a function $$\tilde{f}$$ which is periodic on the larger interval $$x \in [-\Theta, \Theta]$$. We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval $$x \in [-\chi, \chi]$$, identically zero for $$|x| < \Theta$$, and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) $$\mathcal{T}(x; L, \chi, \Theta)=(1+\mbox{erf}(z))/2$$ where $$z=L \xi/\sqrt{1-\xi^{2}}$$ where $$\xi \equiv -1 + 2 (\Theta-x)/(\Theta - \chi)$$. By applying steepest descents to approximate the coefficient integrals in the limit of large degree j, we show that when the width L is fixed, the Fourier cosine coefficients aj of $$\mathcal{T}$$ on $$x \in [-\Theta, \Theta]$$ are proportional to $$a_{j} \sim (1/j) \exp(- L \pi^{1/2} 2^{-1/2} (1-\chi/\Theta)^{1/2} j^{1/2}) \Lambda(j)$$ where Λ(j) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the Nth term, the width should be chosen to increase with N as $$L=0.91 \sqrt{1 - \chi/\Theta} N^{1/2}$$. We derive similar asymptotics for the function f(x)=x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence

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