We consider the problem of determining the asymptotics of the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$. We close this gap and extend the existing lower bounds to all $1\leq p,q\leq \infty$. In the process, we show that the Bernstein widths, which are typically used to lower bound the manifold widths, may decay asymptotically slower than the manifold widths.

翻译：我们研究以$L_p$范数度量误差时Sobolev和Besov空间流形$n$-宽度的渐近行为确定问题。流形宽度控制着这些空间通过一般非线性参数化方法逼近的效率，其约束条件是参数选择与参数化映射必须连续。现有上界与下界仅在Sobolev或Besov光滑指标$q$满足$q\leq p$时匹配。我们填补了这一空白，并将现有下界延伸至所有$1\leq p,q\leq \infty$情形。在此过程中，我们发现通常用于估计流形宽度下界的Bernstein宽度在渐近意义上可能比流形宽度衰减得更慢。