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The proof is not very complicated, but even a sketch needs more space than a comment. So here is a sketch. I want to prove that $$ \int_{-a}^{a} dx \ e^{i y x} \ f(x) = \hat{f}(y) $$ with $f$ and $\hat{f}$ defined as in the question. First one observes that it suffices to prove the equality for $a=1$. Then because of $f(-x)=f(x)$ and the symmetric ...



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