Construction of frames for shift-invariant spaces

Details Construction of frames for shift-invariant spaces Construction of frames for shift-invariant spaces

2011-09-15

arxiv.org/abs/1109.3285v1

Mathematics

Functional Analysis

Scientific paper

We construct a sequence $\{\phi_i(\cdot-j)\mid i=1,...,r,\, j\in{\ZZ}\}$ which constitutes a $p$-frame for the weighted shift-invariant space \[V^p_{\mu}(\Phi)=\Big\{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j) \;\Big|\; \{c_i(j)\}_{j\in{\mathbb{Z}}}\in\ell^p_{\mu},\;i=1,...,r\Big\},\quad p\in[1,\infty],\] 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 gives the Riesz basis.

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