Convex Potentials and Optimal Shift Generated Oblique Duals in Shift Invariant Spaces
We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in L2(Rk). We show that under a natural normalization hypothesis, these convex potentials detec...
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| Autores principales: | , , |
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| Formato: | Articulo |
| Lenguaje: | Inglés |
| Publicado: |
2017
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| Materias: | |
| Acceso en línea: | http://sedici.unlp.edu.ar/handle/10915/96588 https://ri.conicet.gov.ar/11336/66587 https://link.springer.com/article/10.1007%2Fs00041-016-9474-x https://arxiv.org/abs/1508.01739 |
| Aporte de: |
| Sumario: | We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in L2(Rk). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators. |
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