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Diffstat (limited to 'buch/papers/kugel/preliminaries.tex')
| -rw-r--r-- | buch/papers/kugel/preliminaries.tex | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/buch/papers/kugel/preliminaries.tex b/buch/papers/kugel/preliminaries.tex index 1fa78d7..c4c5cae 100644 --- a/buch/papers/kugel/preliminaries.tex +++ b/buch/papers/kugel/preliminaries.tex @@ -288,7 +288,7 @@ way that from now on we will not have to worry about the details of convergence. \begin{lemma} - \label{kugel:lemma:exp-1d} + \label{kugel:thm:exp-1d} The set of functions \(E_n(x) = e^{i2\pi nx}\) on the interval \([0; 1)\) with \(n \in \mathbb{Z} \) are orthonormal. \end{lemma} @@ -318,7 +318,7 @@ convergence. \end{definition} \begin{theorem}[Fourier Theorem] - \label{fourier-theorem-1D} + \label{kugel:thm:fourier-theorem} \begin{equation*} \lim_{N \to \infty} \left \| f(x) - \sum_{n = -N}^N \hat{f}(n) E_n(x) @@ -331,7 +331,7 @@ convergence. on the square \([0; 1)^2\) with \(m, n \in \mathbb{Z} \) are orthonormal. \end{lemma} \begin{proof} - The proof is almost identical to lemma \ref{kugel:lemma:exp-1d}, with the + The proof is almost identical to lemma \ref{kugel:thm:exp-1d}, with the only difference that the inner product is given by \[ \langle E_{m,n}, E_{m', n'} \rangle |