= Fourier decay of repeated interval convolutions
{title2=$|\widehat{1_{[0,1]}^{*m}}(\xi)|\le\min(1,(\pi|\xi|)^{-m})$}
For the <Fourier transform> convention $\widehat f(\xi)=\int f(x)e^{-2\pi i\xi x}\,dx$, integration gives $\widehat{1_{[0,1]}}(\xi)=(1-e^{-2\pi i\xi})/(2\pi i\xi)$, with value one at zero. The <convolution theorem> makes the transform of the $m$-fold <convolution> its $m$th power. This is a simple way to construct compactly supported functions with prescribed polynomial Fourier decay.
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