<p>Radial basis function neural networks (RBFNNs) of Hankel translates are essentially linear combinations of translations and dilations of a so-called activation function defined on the nonnegative real axis, where, instead of the standard translation, the modified Delsarte translation operator associated with the Hankel integral transformation of order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu > -1/2\)</EquationSource> </InlineEquation> is considered. In this paper, we focus on activation functions <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">\(z^{-\mu-1/2}\sigma(z)\)</EquationSource> </InlineEquation> is locally <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-integrable with respect to the measure <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(z^{2 \mu+1} dz\)</EquationSource> </InlineEquation>, for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\le p < \infty\)</EquationSource> </InlineEquation>. It is shown that such networks enjoy the universal approximation property, that is, are locally dense in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-mean if, and only if, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2535_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">\(z^{-\mu-1/2}\sigma(z)\)</EquationSource> </InlineEquation> is not an even polynomial. In this way, a result of Nan, Wu, Long, Ma and Sun (2008) that holds true for RBFNNs of standard translates is extended to RBFNNs of Hankel translates.</p>

Approximation by RBF neural networks of Hankel translates with a locally \(p\)-integrable activation function

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2025年6月16日

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Isabel Marrero

摘要

Radial basis function neural networks (RBFNNs) of Hankel translates are essentially linear combinations of translations and dilations of a so-called activation function defined on the nonnegative real axis, where, instead of the standard translation, the modified Delsarte translation operator associated with the Hankel integral transformation of order \(\mu > -1/2\) is considered. In this paper, we focus on activation functions \(\sigma\) such that \(z^{-\mu-1/2}\sigma(z)\) is locally \(p\) -integrable with respect to the measure \(z^{2 \mu+1} dz\) , for \(1\le p < \infty\) . It is shown that such networks enjoy the universal approximation property, that is, are locally dense in \(p\) -mean if, and only if, \(z^{-\mu-1/2}\sigma(z)\) is not an even polynomial. In this way, a result of Nan, Wu, Long, Ma and Sun (2008) that holds true for RBFNNs of standard translates is extended to RBFNNs of Hankel translates.