What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the largest number of points in $M$ whose distance from each other is $\geq \varepsilon$).

  • Are they are equivalent? If yes, in which sense (i.e. which bounds occur between these three quantities)?
  • Let us suppose that we have a continuous semigroup $(S_t)_{t\ge 0}$ acting on a Banach space $X$ (say, $X=L^p(\mathbb R)$) that gives solutions to a certain PDE. Suppose that we have a bound from above and from below of one of these quantities (say, the $\epsilon$-entropy) of $S_T(C)$ where $C$ is some subset of $X$ and for any $T>0$. Can we use this information to say anything about the convergence properties of a numerical approximation of the PDE?


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