Construction of frames for shift-invariant spaces

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

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