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Shift invariant spaces in \(L^2({\mathbb {R}},{\mathbb {C}}^m)\) with m generators

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Abstract

The paper deals with sampling and reconstruction of vector valued functions in a shift invariant space with multiple generators. Unlike the case of a shift invariant space with multiple generators in \(L^{2}(\mathbb {R})\), when the dimension of the vectors is the same as the number of generators, \(\mathbb {Z}\) turns out to be a stable set of sampling. A sampling formula for reconstructing a function from its samples at integer points is derived and the problem of sampling on a perturbed set of integers is discussed. An illustration of sampling and reconstruction of a function in \(L^{2}(\mathbb {R},\mathbb {R}^{2})\) on a finite interval is given using Matlab.

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Acknowledgements

The authors would like to thank the referee for the valuable suggestions which helped in greatly improving the manuscript. The authors would also like thank Shri. Santi Ranjan Das, Dept. of Mathematics, IIT Madras for important discussions in connection with theorem 4.1.

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Correspondence to R. Radha.

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John, A., Kulkarni, S.H. & Radha, R. Shift invariant spaces in \(L^2({\mathbb {R}},{\mathbb {C}}^m)\) with m generators. J Anal (2020). https://doi.org/10.1007/s41478-019-00219-8

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Keywords

  • Block Laurent operator
  • Reproducing kernel Hilbert space
  • Stable set of sampling
  • Vector valued Zak transform

Mathematics Subject Classification

  • 42C15
  • 94A20