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Sampling and reconstruction in reproducing kernel subspaces of mixed Lebesgue spaces

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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.

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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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  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India

    Anuj Kumar, Dhiraj Patel & Sivananthan Sampath

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  1. Anuj Kumar
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  2. Dhiraj Patel
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  3. Sivananthan Sampath
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Correspondence to Sivananthan Sampath.

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Kumar, A., Patel, D. & Sampath, S. Sampling and reconstruction in reproducing kernel subspaces of mixed Lebesgue spaces. J. Pseudo-Differ. Oper. Appl. 11, 843–868 (2020). https://doi.org/10.1007/s11868-019-00315-0

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