Relaxed conditions for universal approximation by radial basis function neural networks of Hankel translates

Isabel Marrero; Isabel Marrero

doi:10.3934/math.2025493

AIMS Mathematics

2025, Volume 10, Issue 5: 10852-10865. doi: 10.3934/math.2025493

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Relaxed conditions for universal approximation by radial basis function neural networks of Hankel translates

Isabel Marrero ^,

Departamento de Análisis Matemático and Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna (ULL), 38200 La Laguna, Spain

Received: 29 December 2024 Revised: 04 March 2025 Accepted: 06 May 2025 Published: 12 May 2025
MSC : 41A30, 46F12

Radial basis function neural networks (RBFNNs) of Hankel translates of order $ \mu > -1/2 $ with varying widths whose activation function $ \sigma $ is a.e. continuous, such that $ z^{-\mu-1/2}\sigma(z) $ is locally essentially bounded and not an even polynomial, are shown to enjoy the universal approximation property (UAP) in appropriate spaces of continuous and integrable functions. In this way, the requirement that $ \sigma $ be continuous for this kind of networks to achieve the UAP is weakened, and some results that hold true for RBFNNs of standard translates are extended to RBFNNs of Hankel translates.
- universal approximation property,
- RBF,
- nonpolynomiality,
- neural network,
- Hankel translation,
- esssup-norm,
- density,
- activation function
Citation: Isabel Marrero. Relaxed conditions for universal approximation by radial basis function neural networks of Hankel translates[J]. AIMS Mathematics, 2025, 10(5): 10852-10865. doi: 10.3934/math.2025493

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Abstract

Radial basis function neural networks (RBFNNs) of Hankel translates of order $ \mu > -1/2 $ with varying widths whose activation function $ \sigma $ is a.e. continuous, such that $ z^{-\mu-1/2}\sigma(z) $ is locally essentially bounded and not an even polynomial, are shown to enjoy the universal approximation property (UAP) in appropriate spaces of continuous and integrable functions. In this way, the requirement that $ \sigma $ be continuous for this kind of networks to achieve the UAP is weakened, and some results that hold true for RBFNNs of standard translates are extended to RBFNNs of Hankel translates.

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