The Regularity of Refinable Functions

The regularity of refinable functions has been studied extensively in the past. A classical result by Daubechies and Lagarias states that a compactly supported refinable function in $\R$ of finite mask with integer dilation and translations cannot be in $C^\infty$. A bound on the regularity based on the eigenvalues of certain matrices associated with the refinement equation is also given. Surprisingly this fundamental classical result has not been proved in the more general settings, such as in higher dimensions or when the dilation is not an integer. In this paper we extend this classical result to the most general setting for arbitrary dimension, dilation and translations.