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

Abstract

In this work we investigate $n$-widths of multiplier operators ${\Lambda }_{\ast }$ and $\Lambda$, defined for functions on the complex sphere ${\Omega }_d$ of ${\mathbb{C}}^d$, associated with sequences of multipliers of the type $\{{\lambda }_{m,n}^{\ast }{\}}_{m,n\in \mathbb{N}}$, ${\lambda }_{m,n}^{\ast }=\lambda (m+n)$ 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 ${\Lambda }_{\ast }$ and $\Lambda$ are bounded from $L^p({\Omega }_d)$ into $L^q({\Omega }_d)$, $1\le p,q\le \infty$, 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 ${\Lambda }_{\ast }{U}_p$ 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.