Approximations of the Image and Integral Funnel of the $L_p$ Ball under Urysohn Type Integral Operator

TL;DR

This paper develops methods to approximate the image and integral funnel of the $L_p$ ball under Urysohn type integral operators, using finite piecewise-constant functions to achieve internal approximations.

Contribution

It introduces a novel approximation approach for the image and integral funnel of $L_p$ balls under Urysohn operators using discretization with piecewise-constant functions.

Findings

Finite piecewise-constant functions form an internal approximation of the image.

The integral funnel can be approximated by a finite set of points.

Discretization parameters can be chosen to ensure accurate approximations.

Abstract

Approximations of the image and integral funnel of the closed ball of the space $L_p,$ $p>1,$ under Urysohn type integral operator are considered. The closed ball of the space $L_p,$ $p>1,$ is replaced by the set consisting of a finite number of piecewise-constant functions and it is proved that in the appropriate specifying of the discretization parameters, the images of defined piecewise-constant functions form an internal approximation of the image of the closed ball. Applying this result, the integral funnel of the closed ball of the space $L_p,$ $p>1,$ under Urysohn type integral operator is approximated by the set consisting of a finite number of points.