An upper bound on the Kolmogorov widths of a certain family of integral operators

TL;DR

This paper establishes upper bounds on the Kolmogorov widths of a family of integral operators, showing they decrease faster than an exponential of the square root of the dimension, indicating strong approximability.

Contribution

It provides new upper bounds on the Kolmogorov widths for a specific class of integral operators, improving understanding of their approximation properties.

Findings

Kolmogorov widths decrease faster than exp(-κ√n)

Upper bounds are established for the approximation error

Results quantify how well the operators can be approximated by finite-dimensional subspaces

Abstract

We consider the family of integral operators $(K_{\alpha}f)(x)$ from $L^p[0,1]$ to $L^q[0,1]$ given by $$(K_{\alpha}f)(x)=\int_0^1(1-xy)^{\alpha -1}\,f(y)\,\operatorname{d}\!y, \qquad 0<\alpha<1.$$ The main objective is to find upper bounds for the Kolmogorov widths, where the $n$th Kolmogorov width is the infimum of the deviation of $(K_{\alpha}f)$ from an $n$-dimensional subspaces of $L^p[0,1]$ (with the infimum taken over all $n$-dimensional subspaces), and is therefore a measure of how well $K_{\alpha}$ can be approximated. We find upper bounds for the Kolmogorov widths in question that decrease faster than $\exp(-\kappa \sqrt{n})$ for some positive constant $\kappa$.