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author | Nao Pross <np@0hm.ch> | 2022-08-30 22:42:54 +0200 |
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committer | Nao Pross <np@0hm.ch> | 2022-08-30 22:42:54 +0200 |
commit | 1bdb803ced744bcfe7cf81c89a740fcbcf6bdc70 (patch) | |
tree | e2ce322601e737494626056741174ff5acf11e07 /buch/papers/kugel/preliminaries.tex | |
parent | kugel: Minor changes and fix proofs (remove enumerate) (diff) | |
download | SeminarSpezielleFunktionen-1bdb803ced744bcfe7cf81c89a740fcbcf6bdc70.tar.gz SeminarSpezielleFunktionen-1bdb803ced744bcfe7cf81c89a740fcbcf6bdc70.zip |
kugel: Minor corrections
Diffstat (limited to '')
-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 |