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Dann ex. also ein $\delta > 0$ s.d. $\forall \tilde{\delta} \le \delta$ gilt: $\Vert f * \varphi_{\tilde{\delta}} - f \Vert_{L^{1}(\R^{n})} < \frac{\epsilon}{2}$. Setze $K_n \coloneqq \overline{B_n(0)}$ und - $g_n \coloneqq (f * \varphi_{\delta}) \chi_{K_n}$. Dann ist $g_n \in C_c^{\infty}(\R^{n})$, - da $\varphi_{\delta} \in C^{\infty}(\R^{n})$ und $K_n$ kompakt. + $g_n \coloneqq (f * \varphi_{\delta}) \chi_{K_n}$. Dann ist $\text{spt }g_n$ kompakt, + da $\text{spt }g_n = \text{spt } (f * \varphi_{\delta}) \cap \text{spt } \chi_{K_n}$ + als Schnitt zweier abgeschlossener Mengen abgeschlossen und + wegen $\text{spt }\chi_{K_n} = K_n$ und $\text{spt } \chi_{K_n} \supseteq \text{spt }g_n$ beschränkt ist. - Dann gilt $|f|\chi_{K_n} \nearrow |f|$. Also - ex. ein $n_0 \in \N$, s.d. $\forall n \ge n_0$: + Es gilt weiter $|f|\chi_{K_n} \nearrow |f|$. Also + ex. nach dem Satz der monotonen Konvergenz ein $n_0 \in \N$, s.d. $\forall n \ge n_0$: \[ - \left| \Vert f \chi_{K_n} \Vert_{L^{1}} - \Vert f \Vert_{L^{1}} \right| < \frac{\epsilon}{2} + \left| \Vert f \chi_{K_n} \Vert_{L^{1}} - \Vert f \Vert_{L^{1}} \right| < \frac{\epsilon}{3} .\] Damit folgt $\forall n \ge n_0$: \begin{salign*} \underbrace{\Vert f \Vert_{L^{1}}}_{< \infty} &= \underbrace{\int_{K_n}^{} |f| \d{x}}_{< \infty} + \underbrace{\int_{K_n^{c}}^{} |f| \d{x}}_{< \infty} \\ \intertext{Also} \int_{K_n^{c}}^{} |f| \d{x} &= \int_{\R^{n}}^{} |f| \d{x} - \int_{K^{n}}^{} |f| \d{x} \\ - &= \Vert f \Vert_{L^{1}} - \Vert f \chi_{K_n} \Vert \\ - &< \frac{\epsilon}{2} + &= \Vert f \Vert_{L^{1}} - \Vert f \chi_{K_n} \Vert_{L^{1}} \\ + &< \frac{\epsilon}{3} .\end{salign*} - Setze nun $f_{\epsilon} \coloneqq g_{n_0}$. Dann ist $f_{\epsilon} \in C_c^{\infty}$ und es gilt + Der Faltungsapproximationssatz angewendet auf $g_{n_0}$, ergibt ein $\delta' > 0$, s.d. + $\Vert g_{n_0} * \varphi_{\delta '} - g_{n_0} \Vert_{L^{1}} < \frac{\epsilon}{3}$. Nun setze + $f_{\epsilon} \coloneqq g_{n_0} * \varphi_{\delta'}$. Da $\text{spt }g_{n_0}$ kompakt + und $\varphi_{\delta'} \in C_c^{\infty}$ folgt $f_{\epsilon} \in C_c^{\infty}$. + Damit gilt \begin{salign*} - \Vert + \Vert f_\epsilon - f \Vert_{L^{1}} &\le \Vert f_{\epsilon} - g_{n_0} \Vert_{L^{1}} + + \Vert g_{n_0} - f \Vert_{L^{1}} \\ + &= \Vert f_{\epsilon} - g_{n_0} \Vert_{L^{1}} + + \int_{K_{n_0}}^{} |f * \varphi_{\delta} -f | \d{x} + + \int_{K_{n_0}^{c}}^{} |f| \d{x} \\ + &< \frac{\epsilon}{3} + \Vert f * \varphi_{\delta} - f \Vert_{L^{1}} + \frac{\epsilon}{3}\\ + &< \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3}\\ + &= \epsilon .\end{salign*} \end{aufgabe} +\begin{aufgabe}[] + Sei $f \in L^{1}(\R^{n})$ und $\epsilon > 0$. Dann ex. nach Aufgabe 8.2 ein + $f_{\epsilon} \in C_c^{\infty}(\R^{n})$ mit + \[ + \Vert f - f_{\epsilon} \Vert_{L^{1}(\R^{n})} < \frac{\epsilon}{3} \qquad \text{(i)} + .\] Sei nun $|h| < 1$ und setze $S \coloneqq \text{spt}f_{\epsilon}$. Dann ist + $S$ kompakt und beschränkt, d.h. es ex. ein $0 < r < \infty$, s.d. + $\overline{B_r(0)} \supseteq \bigcup_{x \in S} \overline{B_{1}(x)}$. Dann gilt + $\forall x \in \overline{B_r(0)}^{c}\colon f_{\epsilon}(x) = f_{\epsilon}(x + h) = 0$ (ii). + + Da $f_{\epsilon} \in C_c^{\infty}$ ex. nach Lemma von Hadamard $g_i \in C_c^{\infty}(\R^{n})$ mit + \[ + f_{\epsilon}(x) = f_{\epsilon}(0) + \sum_{i=1}^{n} x_i g_i(x) \qquad \forall x \in \R^{n} + .\] Damit gilt $\forall i = 1, \ldots, n$: $|g_i|$ insbesondere stetig und + nimmt damit auf der kompakten Menge $\overline{B_r(0)}$ ein Maximum an. Setze + $M_i \coloneqq \max_{x \in \overline{B_r(0)}} |g_i(x)|$ + und damit $M \coloneqq \max_{i = 1,\ldots,n} M_i$. Sei $M > 0$, sonst + folgt direkt $\Vert f_{\epsilon} - f_{\epsilon, h} \Vert_{L^{1}} = 0$. + + Sei nun $V \coloneqq \mathscr{L}^{n}(\overline{B_r(0)})$. + Dann ist $g_i$ glm. stetig in $\overline{B_r(0)}$, d.h. $\exists \delta_i > 0$, s.d. + $\forall x\in \overline{B_r(0)}$ und $h \in B_{\delta_i}(0)\colon |g_i(x) - g_i(x+h)| < \frac{\epsilon}{6 r n V}$. + Setze nun $\delta \coloneqq \min \{ \delta_i, \frac{\epsilon}{6 n V}\} > 0$. + + Damit folgt für $h \in \R^{n}$ mit $|h| < \min \{\delta, \frac{\delta}{M}\} $: + \begin{salign*} + \Vert f_{\epsilon} - f_{\epsilon, h} \Vert_{L^{1}} + &\stackrel{\text{(ii)}}{=} \int_{\overline{B_r(0)}}^{} |f_{\epsilon}(x) - f_{\epsilon}(x + h)| \d{x} \\ + &\stackrel{\text{Hadamard}}{=} + \int_{\overline{B_r(0)}}^{} \left|f_{\epsilon}(0) + \sum_{i=1}^{n} x_i g_i(x) \d{x} + - f_{\epsilon}(0) - \sum_{i=1}^{n} (x_i + h_i) g_i(x + h) \right| \d{x} \\ + &\le \int_{\overline{B_r(0)}}^{} \sum_{i=1}^{n} |x_i (g_i(x) - g_i(x+h)) - h_i g_i(x +h)| \d{x} \\ + &\le \int_{\overline{B_r(0)}}^{} \sum_{i=1}^{n} (|x_i| |g_i(x) - g_i(x+h)| + |h_i g_i(x +h)| \d{x} \\ + &= \sum_{i=1}^{n} \int_{\overline{B_r(0)}}^{} (\underbrace{|x_i|}_{\le r} + \underbrace{|g_i(x) - g_i(x+h)|}_{< \frac{\epsilon}{6 rnV}} + \underbrace{|h_i|}_{< \frac{\delta}{M}} \underbrace{|g_i(x+h)|}_{\le M} \d{x} \\ + &< \sum_{i=1}^{n} \int_{\overline{B_r(0)}}^{} \Big[ r \frac{\epsilon}{6 r n V} + \frac{\delta}{M} M \Big] \d{x} \\ + &\le \sum_{i=1}^{n} \int_{\overline{B_r(0)}}^{} \left[ \frac{\epsilon}{6 n V} + \frac{\epsilon}{6 n V} \right] \d{\mu} \\ + &= \frac{\epsilon}{3 n V} n V \\ + &= \frac{\epsilon}{3} \qquad \text{(iii)} + .\end{salign*} + Es gilt außerdem $\Vert f_{\epsilon, h} - f_{h} \Vert_{L^{1}} = \Vert f_{\epsilon} - f \Vert_{L^{1}}$ + mit Transformationssatz und der Transformation $z = x + h$ (iv). + Daraus folgt insgesamt + \begin{salign*} + \Vert f - f_h \Vert_{L^{1}} &= \Vert f - f_{\epsilon} + f_{\epsilon} - f_{\epsilon,h} + f_{\epsilon,h} + - f_h \Vert_{L^{1}} \\ + &\le \Vert f - f_{\epsilon} \Vert_{L^{1}} + \Vert f_{\epsilon} - f_{\epsilon,h} \Vert_{L^{1}} + + \Vert f_{\epsilon, h} - f_h \Vert_{L^{1}} \\ + &\stackrel{\text{(iv)}}{=} 2 \Vert f - f_{\epsilon} \Vert_{L^{1}} + \Vert f_{\epsilon} - f_{\epsilon,h} + \Vert_{L^{1}} \\ + &\stackrel{\text{(i), (iii)}}{<} 2 \frac{\epsilon}{3} + \frac{\epsilon}{3} \\ + &= \epsilon + .\end{salign*} + Für $\epsilon \to 0$ folgt die Behauptung. +\end{aufgabe} + \end{document}