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

Abstract 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 $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -spaces, $$1\le p&lt;+\infty ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> Zygmund spaces, exponential spaces and many others.