This problem is a small part of a bigger problem I'm trying to solve. Suppose that $\phi(x) $ is the scaling function for a wavelet orthonormal basis on $L(\mathbb{R})$ and that $P_jf(x) = \sum_k \langle f,\phi_{j,k} \rangle \phi_{j,k} (x)$ is the so-called scale $j$ approximation operator. Given that $f$ is continuous and compactly supported, is it true that this sum consists of only finitlely many terms? Thanks!
(Edit: $\phi_{j,k} = 2^{j/2} \phi (2^jx - k)$ )
If $\phi$ has compact support, then $f$ and $\phi_{j,k}$ have disjoint support except for a finite number of $k$'s and the answer is then yes.