We consider the problem of determining 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 $q\leq p$ or $1 \leq p \leq 2$. We close this gap and obtain sharp lower bounds for all $1 \leq p,q \leq \infty$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $\ell^M_q$-balls in the $\ell_p$-norm when $p\leq q$, which complements existing results that handle the case $q\leq p$. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.

翻译：我们研究了在$L_p$范数误差度量下，索博列夫空间与贝索夫空间的流形$n$宽度确定问题。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，其中参数选择映射与参数化映射必须满足连续性条件。现有上下界仅在索博列夫或贝索夫光滑指数$q$满足$q\leq p$或$1 \leq p \leq 2$时相匹配。我们填补了这一空白，对紧嵌入成立的所有$1 \leq p,q \leq \infty$获得了尖锐下界。分析的关键在于确定有限维$\ell^M_q$球在$\ell_p$范数下流形宽度的精确值（当$p\leq q$时），这补充了现有处理$q\leq p$情形的结果。我们的研究表明，伯恩斯坦宽度（通常用于对流形宽度进行下界估计）在许多情况下具有比流形宽度更快的渐近衰减速度。