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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Tuncer Acar
2024-08-01
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| Series: | Modern Mathematical Methods |
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| Online Access: | https://modernmathmeth.com/index.php/pub/article/view/42 |
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| author | Lorenzo Boccali Danilo Costarelli Gianluca Vinti |
| author_facet | Lorenzo Boccali Danilo Costarelli Gianluca Vinti |
| author_sort | Lorenzo Boccali |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a559b87d90604fa9bd85b81d9bbd65fb |
| institution | DOAJ |
| issn | 3023-5294 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | Tuncer Acar |
| record_format | Article |
| series | Modern Mathematical Methods |
| spelling | doaj-art-a559b87d90604fa9bd85b81d9bbd65fb2025-08-20T03:03:40ZengTuncer AcarModern Mathematical Methods3023-52942024-08-01229010242A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spacesLorenzo Boccali0https://orcid.org/0009-0003-3509-5281Danilo Costarelli1https://orcid.org/0000-0001-8834-8877Gianluca Vinti2https://orcid.org/0000-0002-9875-2790University of FlorenceUniversity of PerugiaUniversity of PerugiaIn 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.https://modernmathmeth.com/index.php/pub/article/view/42neural network operatorsaveraged moduli of smoothnessjackson-type estimatessigmoidal functionshardy-littlewood maximal function |
| spellingShingle | Lorenzo Boccali Danilo Costarelli Gianluca Vinti A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces Modern Mathematical Methods neural network operators averaged moduli of smoothness jackson-type estimates sigmoidal functions hardy-littlewood maximal function |
| title | A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces |
| title_full | A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces |
| title_fullStr | A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces |
| title_full_unstemmed | A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces |
| title_short | A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces |
| title_sort | jackson type estimate in terms of the tau modulus for neural network operators in l p spaces |
| topic | neural network operators averaged moduli of smoothness jackson-type estimates sigmoidal functions hardy-littlewood maximal function |
| url | https://modernmathmeth.com/index.php/pub/article/view/42 |
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