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Let $f \in L^1 (\mathbb R^n)$. Suppose that $f$ is continuous at zero and that the fourier transform $\hat f$ of $f$ is non-negative. Does this imply that $\hat f \in L^1$ (and hence, by the inversion theorem, that $f$ is continuous)? If so, how could I go about proving it?

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    http://math.stackexchange.com/questions/193114/a-positive-fourier-transform-is-integrable?rq=12015-04-30

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