So I am stuck in a proof. The sinc-function serves as synthesis function for the sampling theorem which is defined by : $$sinc(t) \doteqdot \begin{cases}% 1 & \text{if $ t = 0$}\\ \frac{\sin (\pi t)}{\pi t} & \text{if $t\neq 0$} \end{cases}$$ I need to prove that the the integer translates $$t \mapsto sinc(t-k), k \in \mathbb{Z} $$ are pairwise orthogonal with respect to the inner product, i.e.,$$\left\langle sinc (. - k)|sinc (. - l)\right\rangle = \delta_{kl} \enspace for \enspace k,l \in \mathbb{Z} $$ by Using the property that the Fourier transform defines an isometry on $$L^{2}(\mathbb{R} ) , i.e., for \enspace f, g \in L^{2}(\mathbb{R} ) : \left\langle f|g\right\rangle = \left\langle \hat{f}|\hat{g}\right\rangle (Plancherel’s \enspace theorem).$$
Any helps, ideas or any citations to resources to how I can prove this is appreciated.
Thanks.