NEURAL NETWORKS AS POSITIVE LINEAR OPERATORS

This research monograph presents a groundbreaking unification of neural network approximation theory through the lens of Positive Linear Operators (PLOs). For the first time in the literature, neural network operators and activated convolution operators are rigorously analyzed as PLOs — providing a comprehensive, quantitative framework based on inequalities and the modulus of continuity.

The author develops a general, elegant, and highly versatile theory that applies uniformly to a wide variety of neural and convolution operators, bridging Pure and Applied Mathematics with modern Artificial Intelligence and Machine Learning. The results open new directions for mathematical understanding of neural network approximation, with applications across computational analysis, engineering, statistics, and economics.

This volume is an essential resource for mathematicians, computer scientists, and engineers seeking a rigorous analytical foundation for AI and deep learning models.

Contents:

• Neural Networks as Positive Linear Operators
• Neural Networks as Positive Linear Operators over Infinite Domain
• Generalized Logistic Neural Networks Treated as Positive Linear Operators
• Generalized Logistic Neural Networks in Infinite Domain as Positive Linear Operators
• Symmetrized and Perturbed Fuzzy Neural Network Approximation
• Multivariate and Abstract Neural Networks as Positive Linear Operators
• Multivariate Neural Networks in Infinite Domain as Positive Linear Operators and Measure Theory
• Multivariate Generalized Logistic Neural Networks in Infinite Domain as Positive Linear Operators with Measure Theory
• Multivariate Fuzzy-Random Symmetrized and Perturbed Neural Network Strong Approximation
<liMultivariate Generalized Logistic Neural Networks as Positive Linear Operators in Finite Domain
• Approximation by Symmetrized and Disturbed Hyperbolic Tangent Energized Convolutions as Positive Linear Operators
• Approximation by Symmetrized and Disturbed Generalized Logistic Energized Convolutions as Positive Linear Operators
• Complete Approximation by Symmetrized and Disturbed Hyperbolic Tangent Energized Multivariate Convolutions as Positive Linear Operators
• Complete Approximation by Symmetrized and Disturbed Generalized Logistic Energized Multivariate Convolutions in Positive Linear Setting/li>
• Fractional Voronovskaya Asymptotic Expansions by Quasi-Interpolation Neural Network Operators for Brownian Motion
• Neural Network Approximation for Finite Dimension Banach Spaces
• Generalized Logistic Neural Network Approximation in Finite Dimension Banach Spaces

Readership: Researchers and graduate students in applied mathematics, approximation theory, and computational analysis; Academics and professionals in artificial intelligence, machine learning, and deep learning theory; Scientists and engineers in computer science, statistics, data science, and engineering disciplines interested in the mathematical foundations of AI; University and research libraries supporting programs in mathematics, computer science, and engineering.