# Fourier Coefficients of Complex Measure

For my homework I am trying to prove the following:

Suppose $\mu$ is a complex Borel measure on $[0,2\pi)$, and define the Fourier coefficients of $\mu$ by

$\hat{\mu}(n)=\displaystyle\int e^{-int}d\mu(t)$

for $n=0, \pm 1, \pm 2, \ldots$.

Assume that $\hat{\mu}(n)\to 0$ as $n\to +\infty$ and prove that $\hat{\mu}\to 0$ as $n \to -\infty$.

I'm not sure how to get started, can someone provide a hint?

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Use $\lim_{n\to-\infty} \hat{\mu}(n) = \lim_{n\to\infty}\hat{\mu}(-n)$ and find a relation between $e^{-int}$ and $e^{int}$. – Ayman Hourieh Feb 5 '13 at 9:32
@AymanHourieh OK, one is the conjugate of the other. Therefore, $\widehat \mu(-n)=\overline{\widehat {\bar \mu}(n)}$ where $\bar \mu$ is the conjugate of $\mu$. Then how to proceed? – user Aug 15 '13 at 3:04