= Fourier independence from disjoint distribution supports
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Nonzero <compactly supported distributions> with pairwise disjoint supports are linearly independent: a <smooth cutoff function> isolating one support isolates its coefficient. Injectivity transfers this independence to their <Fourier transforms>. Exponential-type bounds and the <Translation property of the Fourier transform> can prove the needed support separation before the distributions themselves are known explicitly.
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