Sampling theorem.

Question

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, h\rangle_{L^2[-\pi,\pi]}=\int_{-\pi}^\pi g(x) \overline{h(x)} dx$. Taking the inverse Fourier transform in (1), we obtain the Whittaker-Kotelnikov-Shannon (WKS) sampling theorem, \begin{equation} f(x)=\sum_{n\in \mathbb Z}f(n) \operatorname{sinc}(x-n), \ \ \ x\in \mathbb R \end{equation} where $f$ is the inverse Fourier transform of the function $\hat{f}$. I would like to know if there is a version of this theorem for $x\in \mathbb C$. Are there any good online resources for it? Thanks!

Answer 1

I am not sure what a "version for $x\in\mathbb{C}$" means, but if you ask if the series representation also holds for all complex $x$, then yes. This is due to the fact that band limited functions are in fact analytic and hence, can be extended to a holomorphic function on the complex plane. Since left and right hand sides of your second equation are both equal on the real line, their unique holomorphic extensions are also equal.