Approximations of the Images and Integral Funnels of the \(L_p\) Balls under a Urysohn-Type Integral Operator

Abstract

Approximations of the image and integral funnel of a closed ball of the space \(L_p\), \(p>1\), under a Urysohn-type integral operator are considered. A closed ball of the space \(L_p\), \(p>1\), is replaced by a set consisting of a finite number of piecewise constant functions, and it is proved that, for appropriate discretization parameters, the images of these piecewise constant functions form an internal approximation of the image of the closed ball. This result is applied to approximate the integral funnel of a closed ball of the space \(L_p\), \(p>1\), under a Urysohn-type integral operator by a set consisting of a finite number of points.