Closure of dilates of shift-invariant subspaces

Let $V$ be any shift-invariant subspace of square summable functions. We prove that if for some $A$ expansive dilation $V$ is $A$-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of $V$, among them the origin is a point of $A^*$-approximate continuity of the spectral function if we assume this value to be one. We present our results also in the more general setting of $A$-reducing spaces. We also prove that the origin is a point of $A^*$-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.