Difference between revisions of "Talk:Aufgaben:Problem 1"
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We can repeat the estimate we did for the first derivative and obviously it results in | We can repeat the estimate we did for the first derivative and obviously it results in | ||
| − | $$ \left| \frac{1}{s_n} \hat | + | $$ \left| \frac{1}{s_n} k \hat g (k) e^{ikx} e^{-i\frac{k^2}{2}t} \left( e^{iks_n} -1 \right) \right| \leq \left| k^2 \cdot \hat g(k) \right| \in \mathcal{S} ({\mathbb{R}}) \subset L^1(\mathbb{R} ) $$ |
Thus we use again dominated convergence and get the second derivative inside. | Thus we use again dominated convergence and get the second derivative inside. | ||
Revision as of 17:02, 14 January 2015
Noch 3 Dinge, aber ansonsten sollte es ok sein:
1)
\(\check{\hat g}(x) = g(x)\)
hier würde ich einfach erwähnen dass dies gilt, da \( g \in S(\mathbb{R}) \)
2)
$$ \left| \frac{1}{s_n} \hat g (k) e^{ikx} e^{-i\frac{k^2}{2}t} \left( e^{i\frac{k^2}{2}s_n} -1 \right) \right|$$
Das müsste meiner Meinung nach um ein Vorzeichen korrigiert werden. Also:
$$ \left| \frac{1}{s_n} \hat g (k) e^{ikx} e^{-i\frac{k^2}{2}t} \left( e^{-i\frac{k^2}{2}s_n} -1 \right) \right|$$
3)
Für die zweite Ableitung nach x wäre mein Vorschlag:
We can repeat the estimate we did for the first derivative and obviously it results in
$$ \left| \frac{1}{s_n} k \hat g (k) e^{ikx} e^{-i\frac{k^2}{2}t} \left( e^{iks_n} -1 \right) \right| \leq \left| k^2 \cdot \hat g(k) \right| \in \mathcal{S} ({\mathbb{R}}) \subset L^1(\mathbb{R} ) $$
Thus we use again dominated convergence and get the second derivative inside.
