A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces
In this paper, we study the order of approximation with respect to the \(L^{p}\)-norm for the (shallow) neural network (NN) operators. We establish a Jackson-type estimate for the considered family of discrete approximation operators using the averaged modulus of smoothness introduced by Sendov and...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Tuncer Acar
2024-08-01
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| Series: | Modern Mathematical Methods |
| Subjects: | |
| Online Access: | https://modernmathmeth.com/index.php/pub/article/view/42 |
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| Summary: | In this paper, we study the order of approximation with respect to the \(L^{p}\)-norm for the (shallow) neural network (NN) operators. We establish a Jackson-type estimate for the considered family of discrete approximation operators using the averaged modulus of smoothness introduced by Sendov and Popov, also known by the name of \(\tau\)-modulus, in the case of bounded and measurable functions on the interval \([-1,1]\). The results here proved, improve those given by Costarelli (J. Approx. Theory 294:105944, 2023), obtaining a sharper approximation. In order to provide quantitative estimates in this context, we first establish an estimate in the case of functions belonging to Sobolev spaces. In the case \(1 < p <+\infty\), a crucial role is played by the so-called Hardy-Littlewood maximal function. The case of \(p=1\) is covered in case of density functions with compact support. |
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| ISSN: | 3023-5294 |