The limits of refinable functions

A function $\phi$ is refinable ( $\phi \in S$) if it is in the closed span of $\{\phi (2x-k)\}$. This set $S$ is not closed in $L_{2}(\mathbb{R})$, and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every $f\in \overline{S} \setminus S$ vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in $[-{\frac{4}{3}}\pi , {\frac{4}{3}}\pi ]$ are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable.