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Sampling and Reconstruction of Signals in a Reproducing Kernel Space with Mixed Norm

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Abstract

In this paper, the signals of interest reside in a reproducing kernel space with mixed norm defined on metric measure spaces. We consider three sampling schemes of the signals of interest depending on the accessible sampling domains that are the whole metric measure spaces or part thereof. We propose iterative algorithms that reconstruct or approximate the original signals from the sampling data. When one can do sampling on the whole metric measure spaces, the proposed algorithm can perfectly reconstruct the original signals if the sampling size is large enough. When one cannot access the whole metric measure spaces, we show that the proposed algorithms can approximate the original concentrated signals with the approximation error being a multiple of the concentration ratio if the sampling set has sufficient density.

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Acknowledgements

The author would like to thank the reviewers for their valuable comments and suggestions. The corresponding author Jun Xian is partially supported by the Guangdong Provincial Government of China through the Computational Science Innovative Research Team program, China; the Guangdong Province Key Laboratory of Computational Science, China; Minjiang Scholars of Fujian province, China.

Funding

The first author Yaxu Li is partially supported by the National Natural Science Foundation of China, China (Grant Nos. (12101169 and 12271140) and the Scientific Research Foundation of Hangzhou Normal University, China (Grant No. 4085C50221204028). The corresponding author Jun Xian is partially supported by the National Natural Science Foundation of China, China (Grant No. 11871481); the Guangdong Province Nature Science Foundation, China (Grant No. 2023A1515012787).

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Authors and Affiliations

  1. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

    Yaxu Li

  2. School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Jun Xian

  3. Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Jun Xian

Authors
  1. Yaxu Li
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  2. Jun Xian
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Correspondence to Jun Xian.

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Li, Y., Xian, J. Sampling and Reconstruction of Signals in a Reproducing Kernel Space with Mixed Norm. Results Math 78, 187 (2023). https://doi.org/10.1007/s00025-023-01965-9

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  • DOI: https://doi.org/10.1007/s00025-023-01965-9

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