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- Jun 20, 2014

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Given a complex Borel measure $\mu$ on the torus $\Bbb T^1$, define the Fourier coefficients of $\mu$ by $\hat{\mu}(n) := \int_{\Bbb T} e^{-2\pi i nx}\, d\mu(x)$, $n\in \Bbb Z$. Show that if the sequence $(\hat{\mu}(n))\in \ell^1(\Bbb Z)$, then $\mu$ has a Radon-Nikyodym derivative with respect to the Lebesgue measure on $\Bbb T$.

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