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-rw-r--r--buch/papers/kugel/preliminaries.tex15
1 files changed, 8 insertions, 7 deletions
diff --git a/buch/papers/kugel/preliminaries.tex b/buch/papers/kugel/preliminaries.tex
index 03cd421..1fa78d7 100644
--- a/buch/papers/kugel/preliminaries.tex
+++ b/buch/papers/kugel/preliminaries.tex
@@ -1,6 +1,6 @@
% vim:ts=2 sw=2 et spell tw=78:
-\section{Preliminaries}
+\section{Preliminaries}\label{kugel:sec:preliminaries}
The purpose of this section is to dust off some concepts that will become
important later on. This will enable us to be able to get a richer and more
@@ -44,23 +44,23 @@ numbers \(\mathbb{R}\).
\)
\end{definition}
-\texttt{TODO: Text here.}
+\kugeltodo{Text here.}
\begin{definition}[Span]
\end{definition}
-\texttt{TODO: Text here.}
+\kugeltodo{Text here.}
\begin{definition}[Linear independence]
\end{definition}
-\texttt{TODO: Text here.}
+\kugeltodo{Text here.}
\begin{definition}[Basis]
\end{definition}
-\texttt{TODO: Text here.}
+\kugeltodo{Text here.}
\begin{definition}[Inner product]
\label{kugel:def:inner-product} \nocite{axler_linear_2014}
@@ -318,11 +318,12 @@ convergence.
\end{definition}
\begin{theorem}[Fourier Theorem]
- \[
+ \label{fourier-theorem-1D}
+ \begin{equation*}
\lim_{N \to \infty} \left \|
f(x) - \sum_{n = -N}^N \hat{f}(n) E_n(x)
\right \|_2 = 0
- \]
+ \end{equation*}
\end{theorem}
\begin{lemma}