In this work we investigate $n$-widths of multiplier operators ${\Lambda }_{\ast }$ and $\Lambda$, defined for functions on the complex sphere ${\Omega }_d$ of ${ℂ}^d$, associated with sequences of multipliers of the type ${\left\{{\lambda }_{m,n}^{\ast }\right\}}_{m,n\in \mathbb{N}}$, ${\lambda }_{m,n}^{\ast }=\lambda (m+n)$ and ${\left\{{\lambda }_{m,n}\right\}}_{m,n\in \mathbb{N}}$, ${\lambda }_{m,n}=\lambda (max\{m,n\})$, respectively, for a bounded function $\lambda$ defined on $[0,\infty )$. If the operators ${\Lambda }_{\ast }$ and $\Lambda$ are bounded from $L^p\left({\Omega }_d\right)$ into $L^q\left({\Omega }_d\right)$, $1\le p,q\le \infty$, and $U_p$ is the closed unit ball of $L^p\left({\Omega }_d\right)$, 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 $L^q\left({\Omega }_d\right)$. 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 $L^q\left({\Omega }_d\right)$, which are order sharp in various important situations.

在这项工作中，我们进行了调查 $ñ$运算符的宽度 ${\Lambda }_{\ast }$ 和 $\Lambda$，为复杂领域的函数定义 ${\Omega }_d$ 的 ${ℂ}^d$，与该类型的乘法器序列相关联 ${\left\{{\lambda }_{米，ñ}^{\ast }\right\}}_{米，ñ\in \mathbb{ñ}}$， ${\lambda }_{米，ñ}^{\ast }=\lambda （米+ñ）$ 和 ${\left\{{\lambda }_{米，ñ}\right\}}_{米，ñ\in \mathbb{ñ}}$， ${\lambda }_{米，ñ}=\lambda （最大限度\{米，ñ\}）$分别用于有界函数 $\lambda$ 定义于 $[0，\infty ）$。如果经营者${\Lambda }_{\ast }$ 和 $\Lambda$ 受限制 ${\mathrm{大号}}^p（{\Omega }_d）$ 进入 ${\mathrm{大号}}^q（{\Omega }_d）$， $\mathrm{1个}\le p，q\le \infty$， 和 ${ü}_p$ 是的闭合单位球 ${\mathrm{大号}}^p（{\Omega }_d）$，我们研究了 $ñ$集的线性Kolmogorov宽度，Gelfand和Bernstein宽度 ${\Lambda }_{\ast }{ü}_p$ 和 $\Lambda {ü}_p$ 在 ${\mathrm{大号}}^q（{\Omega }_d）$。作为应用，我们特别获得了Kolmogorov的估算值$ñ$复球面上具有有限可微，无限可微和解析函数的Sobolev类的宽度 ${\mathrm{大号}}^q（{\Omega }_d）$，在各种重要情况下顺序都很清晰。