Fourier decay of repeated interval convolutions ($|\widehat{1_{[0,1]}^{\ast m}}(\xi )|\le \min (1,(\pi |\xi |{)}^{-m})$)

For the Fourier transform convention $\hat{f}(\xi )=\int f(x)e^{-2\pi i\xi x}dx$, 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.