An upper bound on the Kolmogorov widths of a certain family of integral operators
We consider the family of integral operators $(K_αf)(x)$ from $L^p[0,1]$ to $L^q[0,1]$ given by $$(K_αf)(x)=\int_0^1(1-xy)^{α-1}\,f(y)\,\operatorname{d}\!y, \qquad 0<α<1.$$ 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 $(K_αf)$ 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 $K_α$ can be approximated. We find upper bounds for the Kolmogorov widths in question that decrease faster than $\exp(-κ\sqrt{n})$ for some positive constant $κ$.
math.FA↗