Let $\Phi\in C_c(\mathbb R)$ be a continuous and compactly supported function, satisfying $\int\Phi(x-m)\Phi(x-n)=\delta_{m,n}$. Set $V_j=\mathrm{span}\{\Phi(2^jx-k)\}_{k\in\mathbb Z}$ and consider $\bigcap_{j\leq0}V_j\ni f$, with the understanding that $f\in L^2$. I want to show that $f\equiv0$.
This is easy for some explicitly given $\Phi$. For example, take $\Phi=\mathbf1_{[0,1]}$ equal to the Haar scaling function. Then such an $f$ is constant on $[-2^n,0]$ and $[0,2^n]$ for any $n\geq0$ since $f\in V_{-n}$, and then since $f\in L^2$, we conclude that $f\equiv0$.
For general compactly supported scaling functions we can still say that $f$ is locally only determined by finitely many functions $\phi(2^{j}x-k)$ (for finitely many $k\in\mathbb Z$). However, when I try to write this out, it gets messy quickly.
Is there a clever approach to this? Any help is much appreciated.