On the $L^p$-Convergence and Denoising Performance of Durrmeyer-Type Max-Min Neural Network Operators

TL;DR

This paper studies the convergence and denoising capabilities of Durrmeyer-type max-min neural network operators, demonstrating their effectiveness in smooth approximation and signal filtering through theoretical analysis and numerical examples.

Contribution

It introduces Durrmeyer-type generalizations of max-min neural operators, proving their $L^{p}$ convergence and showcasing their superior denoising performance over existing methods.

Findings

Operators converge in $L^{p}$ norm for functions in $L^{p}([a,b],[0,1])$.

Numerical examples show smoother approximations compared to other operators.

Operators exhibit superior filtering performance in signal analysis.

Abstract

In this paper, we investigate Durrmeyer-type generalizations of maximum-minimum neural network operators. The primary objective of this study is to establish the convergence of these operators in the $L^{p}$ norm for functions $f\in L^{p}([a,b],[0,1])$ with $1\leq p<\infty$. To this end, we analyze the properties of sigmoidal functions and maximum-minimum operations, subsequently establishing the convergence of the proposed operator in pointwise, supremum, and $L^{p}$ norms. Furthermore, we derive quantitative estimates for the rates of convergence. In the applications section, numerical and graphical examples demonstrate that the proposed Durrmeyer-type operators provide smoother approximations compared to Kantorovich-type and standard max-min operators. Finally, we highlight the superior filtering performance of these operators in signal analysis, validating their effectiveness in both approximation and data processing tasks.