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Construction of $p$ frames for weighted shift-invariant spaces
Last modified: 2014-01-31
Abstract
We construct a sequence $\{\phi_i(\cdot-j)\mid j\in{\ZZ},\,i=1,\ldots,r \}$ which constitutes a $p$-frame for
the weighted shift-invariant space
\[V^p_{\mu}(\Phi)=\Big\{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j)
\;\Big|\;
\{c_i(j)\}_{j\in{\mathbb{Z}}}\in\ell^p_{\mu},\;i=1,\ldots,r\Big\},\;
p\in[1,\infty],\] and generates a closed shift-invariant subspace of $L^p_\mu(\mathbb{R})$. The first construction is obtained by choosing functions $\phi_i$, $i=1,\ldots,r$, with compactly supported Fourier transforms $\widehat{\phi}_i$, $i=1,\ldots,r$. The second construction,
with compactly supported $\phi_i,i=1,...,r,$ gives the Riesz basis.
the weighted shift-invariant space
\[V^p_{\mu}(\Phi)=\Big\{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j)
\;\Big|\;
\{c_i(j)\}_{j\in{\mathbb{Z}}}\in\ell^p_{\mu},\;i=1,\ldots,r\Big\},\;
p\in[1,\infty],\] and generates a closed shift-invariant subspace of $L^p_\mu(\mathbb{R})$. The first construction is obtained by choosing functions $\phi_i$, $i=1,\ldots,r$, with compactly supported Fourier transforms $\widehat{\phi}_i$, $i=1,\ldots,r$. The second construction,
with compactly supported $\phi_i,i=1,...,r,$ gives the Riesz basis.
Keywords
$p$-frame; Banach frame; weighted shift-invariant space