Local dual and poly-scale refinability

For a compactly supported function $f$, let $S_n(f), n\ge 0$, be the space of all finite linear combinations of $f(M^n\cdot-k), k\in \mathbf Z$. In this paper, we consider the explicit construction of local duals of $f$ and the poly-scale refinability of functions in $S_0(f)$when $f$ is an $M$-refinable function. We show that for any $M$-refinable function $f$, there exists a local dual of $f$in $S_N(f)$ for some $N\ge 0$, and that any function in $S_0(f)$ is poly-scale refinable.