Frames for weighted shift-invariant spaces

We prove the equivalence of the frame property and the closedness for a weighted shift-invariant space. We also construct a sequence ${\mathrm{\Phi }}^{2k+1}$ and the sequence of spaces $V_{\mu }^p({\mathrm{\Phi }}^{2k+1})$, $k\in \mathbb{N}$, on $\mathbb{R},$ with the useful properties in sampling, approximations and stability.