Approximation of the Image of the Hilbert-Schmidt Integral Operator

TL;DR

This paper presents an approximation method for the image of a ball in Lp space under the Hilbert-Schmidt integral operator, including error estimates, advancing understanding of operator behavior in functional analysis.

Contribution

It introduces a new approximation technique for the image of Lp balls under Hilbert-Schmidt operators with explicit error bounds.

Findings

Derived error estimates for the approximation

Provided explicit bounds for the approximation accuracy

Enhanced understanding of Hilbert-Schmidt operator images in Lp spaces

Abstract

In this paper an approximation of the image of the closed ball of the space $L_p$ $(p>1)$ centered at the origin with radius $r$ under Hilbert-Schmidt integral operator $F(\cdot):L_p\rightarrow L_q$ $\displaystyle \left(\frac{1}{p}+\frac{1}{q}=1\right)$ is presented. An error estimation for given approximation is obtained.