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Approximation Results in Orlicz Spaces for Sequences of Kantorovich Max-Product Neural Network Operators

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Abstract

In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces \(L^{\varphi }\). The results here proved, extend those given by Costarelli and Vinti (Results Math 69(3):505–519, 2016), to a more general context. The main advantage in studying neural network type operators in Orlicz spaces relies in the possibility to approximate not necessarily continuous functions (data) belonging to different function spaces by a unique general approach. Further, in order to derive quantitative estimates in this context, we introduce a suitable K-functional in \(L^{\varphi }\) and use it to provide an upper bound for the approximation error of the above operators. Finally, examples of sigmoidal activation functions have been considered and studied in details.

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Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors are partially supported by the “Department of Mathematics and Computer Science” of the University of Perugia (Italy), and by the GNAMPA-INdAM.

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  1. Department of Mathematics and Computer Science, University of Perugia, 1, Via Vanvitelli, 06123, Perugia, Italy

    Danilo Costarelli & Anna Rita Sambucini

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  1. Danilo Costarelli
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  2. Anna Rita Sambucini
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Correspondence to Danilo Costarelli.

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Costarelli, D., Sambucini, A.R. Approximation Results in Orlicz Spaces for Sequences of Kantorovich Max-Product Neural Network Operators. Results Math 73, 15 (2018). https://doi.org/10.1007/s00025-018-0799-4

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