Abstract
Exploitation of the optimality of (non-exact) frames from a sparse dual point of view is presented. Sparse dual frames and dual Gabor functions of the minimal time and/or frequency supports are studied and constructed through the notion of sparse representations. Conditions on the sparsest dual frames and the dual Gabor functions of the minimal time and/or frequency supports are discussed. Algorithms and examples are provided.
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Acknowledgements
The authors would like to express their gratitude to reviewers for their thoughtful comments that greatly improve the focus and presentation of the article. S. Li is partially supported by the Dean’s Research Fund, School of Information, Renmin University of China, and by NSF grant DMS-1010058.
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Communicated by Akram Aldroubi.
Appendices
Appendix A: Proof of Theorem 3.1
Proof
Since {g m,n } and {γ m,n } are a pair of Gabor frame and dual Gabor frame for ℂd,
We can rewrite (A.1) as, for all t=0,…,d−1,
which is equivalent to, for all t,τ=0,…,d−1,
Substituting expressions of γ m,n (⋅) and g m,n (⋅) into the left hand side, we have, for t,τ=0,…,d−1,
Based on the orthogonality relation
The previous system can be reduced further.
Fix τ∈{0,…,d−1}, for t=0,…,d−1, when (t−τ)modM≠0, (A.2) (A.2) holds automatically. When t=τ, (A.2) is reduced to
When (t−τ)modM=0 and t≠τ, (A.2) is reduced to
A combination of all scenarios above leads to the proposed duality condition. □
Appendix B: Proof of Lemma 4.1
Proof
Given any \(h \in\operatorname{ker}(A)\backslash\{ 0 \}\), based on the support of x, ∥x+h∥1 can be divided into two parts, i.e.,
Since \(\sum_{t \in T} h(t) \operatorname{sgn}( x )(t) + \sum_{t \in T^{C}} |h(t)| > 0\), we see that
Namely, x is the unique solution to (P 1).
For the other direction, assume there exists some \(h_{0} \in \operatorname{ker}(A) \backslash\{ 0 \}\) satisfying
We choose a sufficiently small ϵ>0 such that, for all t∈T,
Then
Since \(\sum_{t\in T} h_{0}(t) \operatorname{sgn}( x )(t) + \sum_{t\in T^{C}} |h_{0}(t)| \leq0\), and ϵ>0,
which is contradictory to x being the unique solution to (P 1). The claim then follows. □
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Li, S., Liu, Y. & Mi, T. Sparse Dual Frames and Dual Gabor Functions of Minimal Time and Frequency Supports. J Fourier Anal Appl 19, 48–76 (2013). https://doi.org/10.1007/s00041-012-9243-4
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DOI: https://doi.org/10.1007/s00041-012-9243-4
Keywords
- Frames
- Sparse dual frames
- Gabor functions
- Sparse dual Gabor functions
- Duality condition
- ℓ 1-Minimization