13
$\begingroup$

Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-Hölder norm at-most $L$ is compact in $C([-1,1]^n,\mathbb{R}^m)$ for the topology of uniform convergence. Therefore, in particular, $X(\alpha,L,r)$ must be totally-bounded.

Are there known lower bounds on the $\varepsilon$-extendal covering number of $X(\alpha,L,r)$; i.e.: the number of uniform $\varepsilon$-balls required to cover $X(\alpha,L)$?

Improve this question
$\endgroup$
11
  • $\begingroup$ do you have any regime in mind? e.g. fixed $\alpha$ and large $L$, or fixed $L$ and small $\alpha$? $\endgroup$
    – mathworker21
    Commented Jul 20, 2021 at 17:59
  • 1
    $\begingroup$ In $1d$ take consecutive points $\{ x_1, \dots, x_n \}$ in $[-1,1]$ whose distance are of order $L^{-1} \epsilon^{1/\alpha}$ so that $n \approx L \epsilon^{-1/\alpha}$. If $|f| \leq 1$ subdivide $[-1,1]$ with points $\{y_1, \dots, y_m\}$ such that $y_{i+1}-y_i \leq \epsilon$ so that $m \approx \epsilon^{-1}$. Given a map $\theta: \{x_1, \dots ,x_n\} \to \{y_1, \dots, y_m\}$ consider $F_\theta$ the set of all functions in $X(\alpha, L)$ with $|f| \leq 1$ such that $f(x_i) \in [\theta(x_i)-\epsilon, \theta(x_i)+\epsilon]$. $\endgroup$
    – Giorgio Metafune
    Commented Jul 20, 2021 at 18:38
  • 1
    $\begingroup$ Thus, $\log N(\varepsilon, X(\alpha, L), \|\cdot\|_{\infty}) \lessapprox m(1/\varepsilon)^{n/\alpha}$ in the general case. $\endgroup$
    – Lars
    Commented Jul 21, 2021 at 6:35
  • 3
    $\begingroup$ Entropies of functional compact sets were studied a lot. (See, for example, doi.org/10.1112/plms/s3-64.1.153 and references.) For Sobolev space the best results are due to Birman and Solomyak. But I believe Holder functions in max-norm should be easier and known for a longer time. $\endgroup$
    – Fedor Petrov
    Commented Jul 21, 2021 at 7:08
  • 1
    $\begingroup$ Thank you to everyone. Each of these are extremely helpful comments/answers. $\endgroup$
    – ABIM
    Commented Aug 1, 2021 at 12:43

0

Reset to default

You must log in to answer this question.