Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ such that $F(y)= \sum_{n\in \mathbb Z}( c_{n}\cdot e^{iny})$, $(y\in \mathbb R).$
Suppose there exists bounded complex Borel measure on $\mathbb R$ such that $$F(y)= \int_{\mathbb R} e^{-iyx} d\mu(x); \ (y\in \mathbb R).$$
My Questions:
(1) What can we say about $\mu$ ? (2) Can we expect to determine $\mu$ just from the above information; or we need to have some more information to determine $\mu$ ? Can you suggests some method to determine $\mu$ in such a situation ?
Thanks,
Observation. Given the representation of $F$ as a Fourier series, a candidate for $\mu$ is $$ \tilde\mu=\sum_{k\in\mathbb Z}c_k\delta_k, $$ where $\delta_k$ is the unit Dirac mass at $k$. Clearly $$ F(y)=\sum_{k\in \mathbb Z}c_k\,\mathrm{e}^{ikx}= \sum_{k\in \mathbb Z}c_k\int_{\mathbb R} \mathrm{e}^{ixy}\,d\delta_k(x)=\int_{\mathbb R} \mathrm{e}^{ixy}\,d\tilde\mu(x). $$
How did I guess that? Simply by considering initially that $F(y)=\mathrm{e}^{kyi}$.
Now we need to show that $\mu=\tilde \mu$. Let $\nu=\mu-\tilde \mu$. Then $$ 0=F(y)-F(y)=\int_{\mathbb R} \mathrm{e}^{ixy}\,d\mu(x)-\int_{\mathbb R} \mathrm{e}^{ixy}\,d\tilde\mu(x)=\int_{\mathbb R} \mathrm{e}^{ixy}\,d\nu(x), \quad \text{for all $y\in\mathbb R.$} $$ But this implies that $\nu=0$.