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

  • Anila John
  • S. H. Kulkarni
  • R. RadhaEmail author
S.I.: ICWAA-2018
  • 8 Downloads

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.

Keywords

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

Mathematics Subject Classification

42C15 94A20 

Notes

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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Copyright information

© Forum D'Analystes, Chennai 2020

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.Naval Physical and Oceanographic LaboratoryDefence Research and Development OrganisationKochiIndia

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