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Sampling and reconstruction in reproducing kernel subspaces of mixed Lebesgue spaces

  • Anuj Kumar
  • Dhiraj Patel
  • Sivananthan SampathEmail author
Article
  • 18 Downloads

Abstract

In this paper, we study the sampling and average sampling problems in a reproducing kernel subspace of mixed Lebesgue space. Let V be an image of \(L^{p,q}({\mathbb {R}}^{d+1})\) under idempotent integral operator defined by a kernel K satisfying certain decay and regularity conditions. Then, we prove that every f in V can be reconstructed uniquely and stably from its samples as well as from its average samples taken on a sufficiently small \(\gamma \)-dense set. Further, we derive iterative reconstruction algorithms for reconstruction of f in V from its samples and average samples. We also obtain the error estimates in iterative reconstruction algorithm from noisy samples and iterative noise.

Keywords

Nonuniform sampling Reproducing kernel subspaces Shift-invariant spaces Average sampling Reconstruction algorithms Integral operator Mixed Lebesgue spaces 

Mathematics Subject Classification

42C15 94A20 47B34 32A70 

Notes

Acknowledgements

The authors are grateful to the anonymous reviewer for meticulously reading the manuscript, and giving us valuable comments and suggestions which helped to improve the quality of the paper.

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Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia

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