Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$.
If $\hat{y}$ is expressed as
\begin{equation} \hat{y}(\omega) = \sum_{k \in \Bbb Z} a_ke^{2\pi i k \omega/\Omega} \end{equation} calculate the coefficients $a_k$.
Show that
\begin{equation} \hat{x} =\begin{cases} \hat{y} & \text{for } |\omega| < \Omega/2\\ 0 & \text{otherwise} \end{cases} \end{equation}
and calculate the values of $x(t)$ by the inverse Fourier Transform.
My Attempt/Input
I know this problem is related to the Whittaker-Kotel'nikov-Shannon Sampling Theorem in some manner. I also know that to calculate the coefficients, I can use the equation
\begin{equation} \frac{1}{\Omega} \int_{-\Omega/2}^{\Omega/2} f(t) e^{-i2\pi k \omega/\Omega} dt \end{equation}
However, I don't quite get what $f(t)$ is here.