Approximation by Steklov Neural Network Operators
S. N. Karaman, M. Turgay, T. Acar
TL;DR
Problem: approximate bounded functions on $[a,b]$ using Steklov Neural Network Operators. Approach: introduce SNNO $F_n^r(f;x)$ defined from Steklov-type integrals $f_{r,h}$ and a sigmoidal-density kernel $phi_sigma$, with $r,n$ in natural numbers and $x$ in $[a,b]$; the case $r=1$ recovers Kantorovich-type NN operators. Contributions: prove well-definedness of $F_n^r$, establish pointwise convergence at continuity points and uniform convergence for $f$ in $C([a,b])$, and obtain a quantitative rate of convergence in terms of the modulus of continuity $omega(f;delta)$, specifically $|(F_n^r f)(x) - f(x)| <= omega(f;n^{-1})/(phi_sigma(r+1)) (1+M_1(phi_sigma))$. Significance: extends neural-network-operator frameworks with Steklov smoothing to broader function classes and provides constructive, quantitative approximation guarantees.
Abstract
The present paper deals with construction of newly family of Neural Network operators, that is, Steklov Neural Network operators. By using Steklov type integral, we introduce a new version of Neural Network operators and we obtain some convergence theorems for the family, such as, pointwise and uniform convergence, rate of convergence via modulus of continuity.