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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 194, Issue 6, pp 1683–1706 | Cite as

Sampling and reconstruction in shift-invariant spaces on \(\mathbb {R}^d\)

  • A. Antony Selvan
  • R. Radha
Article

Abstract

Let \(\phi \in W(C,\ell ^1)\) such that \(\{\tau _n\phi :n\in \mathbb {Z}^d\}\) forms a Riesz basis for \(V(\phi )\). It is shown that \(\mathbb {Z}^d\) is a stable set of sampling for \(V(\phi )\) if and only if \(\Phi ^\dagger (x)\ne 0\), for every \(x\in \mathbb {T}^d\), where \(\Phi ^{\dagger }(x):=\sum _{n\in \mathbb {Z}^d}\phi (n)e^{2\pi in\cdot x},~~ x\in \mathbb {T}^d\). Sampling formulae are provided for reconstructing a function \(f\in V(\phi )\) from uniform samples using Zak transform and complex analytic technique. The problem of sampling and reconstruction is discussed in the case of irregular samples also. The theory is illustrated with some examples, and numerical implementation for reconstruction of a function from its nonuniform samples is provided using MATLAB.

Keywords

Frames Laurent operator Riesz basis Shift-invariant space Wiener amalgam space Zak transform 

Mathematics Subject Classification

94A20 42C15 42B99 

Notes

Acknowledgments

We wish to thank the referee for meticulously reading the manuscript and giving us valuable suggestions. In fact, we owe the revised version completely to the referee as he made us realize the fact that our results are valid with a weakened hypothesis.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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