Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators

Danilo Costarelli; Gianluca Vinti

doi:10.3934/mfc.2020004

Article Contents

2020, Volume 3, Issue 1: 41-50. Doi: 10.3934/mfc.2020004

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Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators

Danilo Costarelli^, and
Gianluca Vinti

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

^* Corresponding author: Danilo Costarelli
^* Corresponding author: Danilo Costarelli

Received: December 2019

Revised: February 2020

Published: February 2020

The first author has been partially supported within the 2019 GNAMPA-INdAM Project "Metodi di analisi reale per l'approssimazione attraverso operatori discreti e applicazioni", while the second author within the projects: (1) Ricerca di Base 2018 dell'Università degli Studi di Perugia - "Metodi di Teoria dell'Approssimazione, Analisi Reale, Analisi Nonlineare e loro Applicazioni", (2) Ricerca di Base 2019 dell'Università degli Studi di Perugia - "Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni", (3) "Metodi e processi innovativi per lo sviluppo di una banca di immagini mediche per fini diagnostici" funded by the Fondazione Cassa di Risparmio di Perugia, 2018

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Abstract

In this paper, an asymptotic formula for the so-called multivariate neural network (NN) operators has been established. As a direct consequence, a first and a second order pointwise Voronovskaja type theorem has been reached. At the end, the particular case of the NN operators activated by the logistic function has been treated in details.

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Mathematics Subject Classification: Primary: 41A25, 41A05; Secondary: 41A30, 47A58.

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Figure 1. The function $ \phi_{\sigma_{\ell}} $

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Figure 2. The function $ \Psi_{\sigma_{\ell}} $ of two variables

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