Masterarbeit/TeX/theo_3_8.tex

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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.