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Random Average Sampling in a Reproducing Kernel Subspace of Mixed Lebesgue Space \(L^{p,q}({\mathbb {R}}^{n+1})\)

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Abstract

In this paper, we study average sampling inequality in a probabilistic framework for a reproducing kernel subspace V of mixed Lebesgue space. More precisely, we show with high probability that a function concentrated on a compact cube C can be stably recovered from their \({\mathcal {O}}(\mu (C)\log \mu (C))\) many average values at uniformly distributed random points over C, where \(\mu \) is a Lebesgue measure. Further, we propose an exponential convergence reconstruction scheme to reconstruct the concentrated function from their random average measurements and illustrate with an example.

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Acknowledgements

We are grateful to the anonymous referee for meticulously reading the manuscript and giving us helpful suggestions. The authors thank IIT Delhi HPC facility for computational resources. The first author acknowledges Council of Scientific & Industrial Research for financial support. The second author is grateful for the financial assistance received under project number CRG/2019/002412 from the Department of Science and Technology, Government of India.

Funding

The work is supported by the Council of Scientific & Industrial Research, India, and the Department of Science and Technology, Government of India.

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

    Dhiraj Patel & S. Sivananthan

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  1. Dhiraj Patel
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  2. S. Sivananthan
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Correspondence to Dhiraj Patel.

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Patel, D., Sivananthan, S. Random Average Sampling in a Reproducing Kernel Subspace of Mixed Lebesgue Space \(L^{p,q}({\mathbb {R}}^{n+1})\). Results Math 79, 9 (2024). https://doi.org/10.1007/s00025-023-02037-8

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