Approximation by Neural Network Sampling Operators in Mixed Lebesgue Spaces
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909579065950208 |
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| author | Dey, Arpan Kumar Kumar, A. Sathish Devaraj, P. |
| author_facet | Dey, Arpan Kumar Kumar, A. Sathish Devaraj, P. |
| contents | In this paper, we prove the rate of approximation for the Neural Network Sampling Operators activated by sigmoidal functions with mixed Lebesgue norm in terms of averaged modulus of smoothness for a bounded measurable functions on bounded domain. In order to achieve the above result, we first establish that the averaged modulus of smoothness is finite for certain suitable subspaces of $L^{p,q}(\mathbb{R}\times\mathbb{R}).$ Using the properties of averaged modulus of smoothness, we estimate the rate of approximation of certain linear operators in mixed Lebesgue norm. Then, as an application of these linear operators, we obtain the Jackson type approximation theorem, in order to give a characterization for the rate of approximation of neural network operators in-terms of averaged modulus of smoothness in mixed norm. Lastly, we discuss some examples of sigmoidal functions and using these sigmoidal functions, we show the implementation of continuous and discontinuous functions by neural network operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximation by Neural Network Sampling Operators in Mixed Lebesgue Spaces Dey, Arpan Kumar Kumar, A. Sathish Devaraj, P. Functional Analysis In this paper, we prove the rate of approximation for the Neural Network Sampling Operators activated by sigmoidal functions with mixed Lebesgue norm in terms of averaged modulus of smoothness for a bounded measurable functions on bounded domain. In order to achieve the above result, we first establish that the averaged modulus of smoothness is finite for certain suitable subspaces of $L^{p,q}(\mathbb{R}\times\mathbb{R}).$ Using the properties of averaged modulus of smoothness, we estimate the rate of approximation of certain linear operators in mixed Lebesgue norm. Then, as an application of these linear operators, we obtain the Jackson type approximation theorem, in order to give a characterization for the rate of approximation of neural network operators in-terms of averaged modulus of smoothness in mixed norm. Lastly, we discuss some examples of sigmoidal functions and using these sigmoidal functions, we show the implementation of continuous and discontinuous functions by neural network operators. |
| title | Approximation by Neural Network Sampling Operators in Mixed Lebesgue Spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2504.10206 |