# A Sampling theorem for power signals

Any function $f \in PW_{\pi}$ (where $PW_{\pi}$ is the Paley-Wiener space), can be expanded in terms of the orthonormal basis $\{e^{i n x}\}_{n\in \mathbb Z}$ as $$\hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2(-\pi,\pi)} e^{i n x}$$ where $\langle g, h\rangle_{L^2(-\pi,\pi)}=\frac{1}{2\pi}\int_{-\pi}^\pi g(x) \overline{h(x)} dx$. Taking the inverse Fourier transform in previous equation, we obtain the Whittaker-Kotelnikov-Shannon (WKS) sampling theorem, $$f(x)=\sum_{n\in \mathbb Z}f(n) \operatorname{sinc}(x-n), \ \ \ x\in \mathbb R$$ where $\operatorname{sinc}(x)$ is the normalized sinc function. My question is the following. Sampling theorem, at least in the version that I wrote above, is related to $L^2$ space and so, to scalar product: $$\langle g, h\rangle_{L^2(-\pi,\pi)}=\frac{1}{2\pi}\int_{-\pi}^\pi g(x) \overline{h(x)} dx$$ Instead, usually, for power signals, we define the inner product to be $$\left\langle g\,,\,h\right\rangle = \lim_{T\rightarrow \infty} \frac 1 {2T} \int_{-T}^T g(x)\overline{h(x)}\,dx \tag{1}$$ Is there a version of Sampling theorem related to scalar product (1)? Is it used in signal processing?