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<p>In this paper, we introduce a new family of neural network (NN) operators, namely the Steklov-type max-product NN operators. These operators extend the use of max-product techniques in function approximation by incorporating Steklov-type integrals. As it is well known, NN operators are typically generated by several kinds of sigmoidal activation functions, such as the ramp function, whose density functions can be expressed in terms of ReLU (Rectified Linear Unit). In particular, we establish pointwise and uniform convergence theorems for this class of operators in the space of continuous functions, and extend our analysis to the broader setting of Orlicz spaces, which include not necessarily continuous functions. The framework of Orlicz spaces further allows us to deduce convergence results in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spaces, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(1\le p&lt;+\infty ,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>&lt;</mo> <mo>+</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> Zygmund spaces, exponential spaces and many others.</p>

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Uniform and modular convergence of max-product neural network operators of Steklov-type

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月17日

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  • Mariarosaria Natale,
  • Michele Piconi

摘要

In this paper, we introduce a new family of neural network (NN) operators, namely the Steklov-type max-product NN operators. These operators extend the use of max-product techniques in function approximation by incorporating Steklov-type integrals. As it is well known, NN operators are typically generated by several kinds of sigmoidal activation functions, such as the ramp function, whose density functions can be expressed in terms of ReLU (Rectified Linear Unit). In particular, we establish pointwise and uniform convergence theorems for this class of operators in the space of continuous functions, and extend our analysis to the broader setting of Orlicz spaces, which include not necessarily continuous functions. The framework of Orlicz spaces further allows us to deduce convergence results in \(L^p\) L p -spaces, \(1\le p<+\infty ,\) 1 ≤ p < + ∞ , Zygmund spaces, exponential spaces and many others.