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Generalized Sampling in \({L}^{2}({\mathbb{R}}^{d})\)Shift-Invariant Subspaces with Multiple Stable Generators

  • H. R. Fernández-Morales
  • A. G. García
  • G. Pérez-Villalón
Chapter

Abstract

In order to avoid most of the problems associated with classical Shannon’s sampling theory, nowadays, signals are assumed to belong to some shift-invariant subspace. In this work we consider a general shift-invariant space V Φ 2 of L 2(ℝ d ) with a set Φ of r stable generators. Besides, in many common situations, the available data of a signal are samples of some filtered versions of the signal itself taken at a sub-lattice of ℤ d . This leads to the problem of generalized sampling in shift-invariant spaces. Assuming that the 2-norm of the generalized samples of any fV Φ 2 is stable with respect to the L 2(ℝ d )-norm of the signal f, we derive frame expansions in the shift-invariant subspace allowing the recovery of the signals in V Φ 2 from the available data. The mathematical technique used here mimics the Fourier duality technique which works for classical Paley–Wiener spaces.

Keywords

Filter Bank Separable Hilbert Space Riesz Basis Wiener Space Reconstruction Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The authors are very pleased to dedicate this work to Professor Gilbert. G.Walter on the occasion of his 80th birthday. Professor Walter’s research andmentorship have, over the years, inspired and influenced many mathematiciansthroughout the world; we are fortunate to be three of these mathematicians.

This work has been supported by the grant MTM2009–08345 from the Spanish Ministerio de Ciencia e Innovación (MICINN).

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Copyright information

© Springer New York 2013

Authors and Affiliations

  • H. R. Fernández-Morales
    • 1
  • A. G. García
    • 1
  • G. Pérez-Villalón
    • 2
  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés (Madrid)Spain
  2. 2.Departamento de Matemática AplicadaE.U.I.T.T., U.P.M.MadridSpain

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