I'm having some issues with the following problem. The problem consists of having knowledge of the Paley-Wiener space
$$PW:=\{f\in L^{2}(\mathbb{R})\ \big{|}\ \operatorname{supp}\ \hat{f}\subseteq [-\frac{1}{2},\frac{1}{2}]\}$$
where $\hat{f}$ denotes the Fourier transform of $f$. Shannon's sampling theorem says if a function $f\in PW$ then
$$f(x)=\sum_{k\in \mathbb{Z}}f(k)\operatorname{sinc}(x-k)$$
where $\operatorname{sinc}(x)=\displaystyle \frac{\sin(\pi x)}{\pi x}$ if $x\neq 0$ and $\operatorname{sinc}(x)=1$ if $x=0$.
The problem begins here:
Let $f\in L^{2}(\mathbb{R})$ be a continuous function for which $\operatorname{supp}\ \hat{f}\subseteq [-\alpha/2,\alpha/2]$ for some $\alpha>0$. Show that $f$ can be recovered form its samples $\{f(k/\alpha)\}_{k\in \mathbb{Z}}$ via
$$f(x)=\sum_{k\in \mathbb{Z}}f(\frac{k}{\alpha})\operatorname{sinc}(\alpha x-k),\ x\in \mathbb{R}$$
Here's where I've gotten so far:
If we let $g\in PW$ then we know from Shannon's sampling theorem that
$$g(x)=\sum_{k\in \mathbb{Z}}g(k)\operatorname{sinc}(x-k)$$
We can then consider the function $f\in L^{2}(\mathbb{R})$ as
$$f=D_{\alpha}g$$
where $D_{\alpha}$ is the dilation operator. This means that $\hat{f}$ has $\operatorname{supp}\ \hat{f}\subseteq [-\alpha/2,\alpha/2]$. My idea is then to apply Shannon's sampling theorem to $D_{\alpha}g$, but I'm not sure if that is justified, since $f$ has to be in $PW$, and I'm not sure if $D_{\alpha}g$ is in $PW$. Furthermore, if I choose to apply Shannon's sampling theorem, I get
$$f(x)=(D_{\alpha}g)(x)=\alpha^{-\frac{1}{2}}g(\frac{x}{\alpha})=\sum_{k\in \mathbb{Z}}\alpha^{-\frac{1}{2}}g(\frac{k}{\alpha})\operatorname{sinc}(x-k)$$
But I'm not sure I'm doing it correctly. I would appreciate any hints on how to get further with this problem. Thanks in advance!