On the density of translation networks defined on the unit ball

Xiaojun Sun; Baohuai Sheng; Lin Liu; Xiaoling Pan

doi:10.3934/mfc.2023017

Article Contents

2024, Volume 7, Issue 3: 386-404. Doi: 10.3934/mfc.2023017

This issue Previous Article Complex network pinning control based On DR algorithm Next Article Asymptotically deferred statistical equivalent functions of order $ \alpha $ in amenable semigroups

On the density of translation networks defined on the unit ball

1.
Department of Economic Statistics, School of International Business, Zhejiang Yuexiu University, Shaoxing 312000 China
2.
School of Mathematical Physics and Information, Shaoxing University, Shaoxing, Zhejiang 312000, China

^*Corresponding author: Baohuai Sheng
^*Corresponding author: Baohuai Sheng

Received: December 2022

Revised: January 2023

Early access: March 2023

Published: August 2024

This research is supported by the NSFC/RGC Joint Research Scheme (Project No. 12061160462 and N_CityU 102/20) and the NSF (Project No. 61877039) of China

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Abstract

We call a translation function class a set of zonal translation networks if it is produced by the translations of a convolutional structure, which is a composite function reunited with an even function defined on $ [-1, 1] $ and a geodesic distance. In the present paper, we consider the density of a class of zonal translation networks produced by a convolution structure on the unit ball from the view of Fourier-Laplace series and give a sufficient condition to ensure the zonal translation class is density in $ L^p $ spaces defined on the unit ball. In particular, we construct with De La Vallée Poussin operators a sequence of zonal translation networks and show the approximation order.

Keywords:

Convolutional structure on the unit ball,
Jacobi orthogonal polynomials,
zonal translations,
Fourier-Laplace series,
density.

Mathematics Subject Classification: 68Q32, 68T05.

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