# On Approximation Properties for Non-linear Integral Operators

## TL;DR

This paper studies the pointwise convergence of a family of non-linear integral operators with non-negative kernels, applicable on finite intervals and the entire real line, contributing to the understanding of their approximation properties.

## Contribution

It provides new results on the convergence behavior of non-linear integral operators with parameter-dependent kernels for functions in L1 spaces.

## Key findings

- Established conditions for pointwise convergence on finite intervals.
- Extended convergence results to the whole real line.
- Analyzed the impact of kernel properties on approximation accuracy.

## Abstract

We investigate the problem of pointwise convergence of the family of non-linear integral operators: \begin{equation} L_\lambda(f,x) = \int_a^b \sum_{m=1}^N f^m(t) K_{\lambda ,m}(x,t) dt, \end{equation} where $\lambda $ is a real parameters, $K_{\lambda ,m}(x,t)$ is non-negative kernel and $f$ is the function in $L_{1}(a,b)$. We consider two cases where $% (a,b)$ is a finite interval and when is the whole real axis.