%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:

With the following Theorem we will have an explicit desrctiption for the
limes of RN as the amount of nodes is increased.

\begin{Theorem}[Ridge weight penaltiy corresponds to adapted spline]
  \label{theo:main1}
  For arbitrary training data \(\left(x_i^{train}, y_i^{train}\right)\) it holds
  \[
    \plimn \norm{\mathcal{RN^{*, \tilde{\lambda}}} - f^{*,
        \tilde{\lambda}}_{g, \pm}}_{W^{1,\infty}(K)} = 0.
  \]

  With
  \begin{align*}
    \label{eq:1}
    \tilde{\lambda} &\coloneqq \lambda n g(0), \\
    g(x) &\coloneqq
           g_{\xi}(x)\mathbb{E}\left[ v_k^2 \vert \xi_k = x \right], \forall x
           \in \mathbb{R}
  \end{align*}
  and \(RN^{*, \tilde{\lambda}}\), \(f^{*,\tilde{\lambda}}_{g, \pm}\)
  as defined in ??? and ??? respectively. 
\end{Theorem}
In order to proof Theo~\ref{theo:main1} we need to proof a number of
auxilary Lemmata first.

\begin{Definition}[Sobolev Norm]
  \label{def:sobonorm}
  The natural norm of the sobolev space is given by
  \[
    \norm{f}_{W^{k,p}(K)} =
    \begin{cases}
      \left(\sum_{\abs{\alpha} \leq k}
        \norm{f^{(\alpha)}}^p_{L^p}\right)^{\nicefrac{1}{p}},&
      \text{for } 1 \leq p < \infty \\
      max_{\abs{\alpha} \leq k}\left\{f^{(\alpha)}\right\},& \text{for
      } p = \infty
    \end{cases}
    .
  \]
\end{Definition}

\begin{Lemma}[Poincar\'e typed inequality]
  Let \(f:\mathbb{R} \to \mathbb{R}\) differentiable with \(f' :
  \mathbb{R} \to \mathbb{R}\) Lesbeque integrable. Then for \(K=[a,b]
  \subset \mathbb{R}\) with \(f(a)=0\) it holds that
  \begin{equation}
    \label{eq:pti1}
    \exists C_K^{\infty} \in \mathbb{R}_{>0} :
    \norm{f}_{w^{1,\infty}(K)} \leq C_K^{\infty}
    \norm{f'}_{L^{\infty}(K)}.
  \end{equation}
  If additionaly \(f'\) is differentiable with \(f'': \mathbb{R} \to
  \mathbb{R}\) Lesbeque integrable then additionally
  \begin{equation}
    \label{eq:pti2}
    \exists C_K^2 \in \mathbb{R}_{>0} : \norm{f}_{W^{1,\infty}(K)} \leq
    C_K^2 \norm{f''}_{L^2(K)}.
  \end{equation}
  \proof
  With the fundamental theorem of calculus, if
  \(\norm{f}_{L^{\infty}(K)}<\infty\) we get
  \begin{equation}
    \label{eq:f_f'}
    \norm{f}_{L^{\infty}(K)} = \sup_{x \in K}\abs{\int_a^x f'(s) ds} \leq
    \sup_{x \in K}\abs{\int_a^x \sup_{y \in K} \abs{f'(y)} ds} \leq \abs{b-a}
    \sup_{y \in K}\abs{f'(y)}.
  \end{equation}
  Using this we can bound \(\norm{f}_{w^{1,\infty}(K)}\) by
  \[
    \norm{f}_{w^{1,\infty}(K)} \stackrel{\text{Def~\ref{def:sobonorm}}}{=}
    \max\left\{\norm{f}_{L^{\infty}(K)},
      \norm{f'}_{L^{\infty}(K)}\right\}
    \stackrel{(\ref{eq:f_f'})}{\leq} max\left\{\abs{b-a},
      1\right\}\norm{f'}_{L^{\infty}(K)}.
  \]
  With \(C_k^{\infty} \coloneqq max\left\{\abs{b-a}, 1\right\}\) we
  get (\ref{eq:pti1}).
  By using the Hölder inequality, we can proof the second claim.
  \begin{align*}
    \norm{f'}_{L^{\infty}(K)} &= \sup_{x \in K} \abs{\int_a^bf''(y)
      \mathds{1}_{[a,x]}(y)dy} \leq \sup_{x \in
      K}\norm{f''\mathds{1}_{[a,x]}}_{L^1(K)}\\
                              &\hspace{-6pt} \stackrel{\text{Hölder}}{\leq} sup_{x
                                \in
                                K}\norm{f''}_{L^2(K)}\norm{\mathds{1}_{[a,x]}}_{L^2(K)}
                                = \abs{b-a}\norm{f''}_{L^2(K)}.
  \end{align*}
  Thus (\ref{eq:pti2}) follows with \(C_K^2 \coloneqq
  \abs{b-a}C_K^{\infty}\).
  \qed
\end{Lemma}