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Construction of frames for shift-invariant spaces
Functional Analysis · Submitted 15 Sep 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $L^p_\mu(\mathbb{R})$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $L^p_\mu(\mathbb{R})$. | contribution 0.93 | 0.94 | 0.81 | asserted 1.00 |
| s1 | The first construction is obtained by choosing functions $\phi_i$, $i=1,...,r$, with compactly supported Fourier transforms $\hat{\phi}_i$, $i=1,...,r$. | method 0.95 | 0.87 | 0.08 | asserted 1.00 |
| s2 | The second construction, with compactly supported $\phi_i,i=1,...,r,$ gives the Riesz basis. | result 0.72 | 0.88 | 0.15 | asserted 1.00 |
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Original 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.
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Contests 1 paper
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These results expand and correct the construction obtained in [ref] .
Frames for weighted shift-invariant spaces · Introduction and preliminaries · contest 0.88
Cites as background 1 paper
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We will recall some results from [ref] and [ref] which are needed in the sequel.
Frames for weighted shift-invariant spaces · Basic spaces · 0.94
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