Construction of frames for shift-invariant spaces

Abstract: We construct a sequence ${\phi_i(\cdot-j)\mid j\in{\ZZ}, i=1,...,r}$ which constitutes a $p$-frame for the weighted shift-invariant space [V^p_{\mu}(\Phi)=\Big{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j) \Big| {c_i(j)}{j\in{\mathbb{Z}}}\in\ell^p{\mu}, i=1,...,r\Big}, p\in[1,\infty],] and generates a closed shift-invariant subspace of $L^p_\mu(\mathbb{R})$. The first construction is obtained by choosing functions $\phi_i$, $i=1,...,r$, with compactly supported Fourier transforms $\hat{\phi}_i$, $i=1,...,r$. The second construction, with compactly supported $\phi_i,i=1,...,r,$ gives the Riesz basis.