Estimates for $n$-widths of sets of smooth functions on complex spheres
In this work we investigate $n$-widths of multiplier operators $Λ_*$ and $Λ$, defined for functions on the complex sphere $Ω_d$ of $\mathbb{C}^d$, associated with sequences of multipliers of the type $\{λ_{m,n}^*\}_{m,n\in \mathbb{N}}$, $λ_{m,n}^*=λ(m+n)$ and $\{λ_{m,n}\}_{m,n\in \mathbb{N}}$, $λ_{m,n}=λ(\max\{m,n\})$, respectively, for a bounded function $λ$ defined on $[0,\infty)$. If the operators $Λ_{*}$ and $Λ$ are bounded from $L^p(Ω_d)$ into $L^q(Ω_d)$, $1\leq p,q\leq\infty$, and $U_p$ is the closed unit ball of $L^p(Ω_d)$, we study lower and upper estimates for the $n$-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of the sets $Λ_{*}U_p$ and $ΛU_p$ in $L^q(Ω_d)$. 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 $L^q(Ω_d)$, which are order sharp in various important situations.