Estimates for $n$-widths of sets of smooth functions on complex spheres

Abstract

In this work we investigate $n$-widths of multiplier operators $\Lambda_*$ and $\Lambda$, defined for functions on the complex sphere $\Omega_d$ of $\mathbb{C}^d$, associated with sequences of multipliers of the type $\{\lambda_{m,n}^*\}_{m,n\in \mathbb{N}}$, $\lambda_{m,n}^*=\lambda(m+n)$ and $\{\lambda_{m,n}\}_{m,n\in \mathbb{N}}$, $\lambda_{m,n}=\lambda(\max\{m,n\})$, respectively, for a bounded function $\lambda$ defined on $[0,\infty)$. If the operators $\Lambda_{*}$ and $\Lambda$ are bounded from $L^p(\Omega_d)$ into $L^q(\Omega_d)$, $1\leq p,q\leq\infty$, and $U_p$ is the closed unit ball of $L^p(\Omega_d)$, we study lower and upper estimates for the $n$-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of the sets $\Lambda_{*}U_p$ and $\Lambda U_p$ in $L^q(\Omega_d)$. As application we obtain, in particular, estimates for the Kolmogorov $n$-width of classes of Sobolev, of finitely differentiable, infinitely differentiable and analytic functions on the complex sphere, in $L^q(\Omega_d)$, which are order sharp in various important situations.