What kind of functions, $f \in L^2(\mathbb{R})$ are such that $\operatorname{supp} \hat{f} \subset [-\pi,\pi]$?
I am trying to prove that the set $\{V_j\}_{j \in \mathbb{Z}}$ consisting of functions $f \in L^2(\mathbb{R})$ such that $\operatorname{supp} \hat{f} \subset [-2^j\pi,2^j\pi]$ is a multiresolution.
I have proven every property except for finding the scaling function $\phi$ such that $\phi (\cdot -k)$ forms a Riesz Basis of $V_0$
Since $V_0$ consists of functions, $f \in L^2(\mathbb{R})$ such that $supp \hat{f} \subset [-\pi,\pi]$ I am trying to get an idea of what $V_0$ "looks like" in order to find this Riesz Basis. Any ideas or hints would be greatly appreciated.