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zZxE26$@y^;!E+@_jDBNR3T0HRa?fZ0TXmLrK2MwwmHW)0;zA{@b%o)4Pl4 z?^(nNdt&j!@M~R{jvS>`T&v~>|3mh}!~Ws9k&?c=diiTnS8Msv!+TX%FjwU-r2c5a zv$b!H^Oh3ZYO}ho%!TH1&Caa#h3r%L`q%8&EpLxB;p3ONlhM7cELPqNHHWxD$@zH- zA9Bl~4WqacPEw)2s7-%%fdpNl0_1%?9A6oTTda^xz;>j)@WBGt0YqR2ze&mHNazS8 znd$`AX`$&AIi7uDMxQ?pPR5l}{;0PnA-^EFV*?nCyvp%))+Zj7@}C-Kv!0PmI?4#b`9;R2GV%-dF8he)t0NPZhiUj%Iv|Fge6I+(UxF79EDq zF0OX7f^?z2bm_Z62XFSw3I}Fi&Dhk`%1^R!Pt|PMB)1an+om7gwCqLUxM_yQnYWhd zz8Mq0&U`hf3(L}1GLCt&_3F6wb>-F6A9h`S@VDoWR@Rq-RIv?#vywk($T655&41bO zy;-&C$X30bb4!$Vd*8h8hyMDTbs-(cbCVwiG}i+ec1z1N+?M&*qI4kT@#cpF?bp*@ zwl9zSu-gTTS`t!Ve-T|2U|}hy&8>UhG&E!2hzS{6s?Wh??#?WcO9GaV|8wttV*E<4 z!{^p$DXw7TWW?Y~NLd6|^<<*WzbPEw-erjEz|yJh>i$Ek+tMh6PY&{AE1AFB&G9zC zezn&YrYD+C6MgV2H~^{Wewxrvc^xoU&PM=QI(}rk)%kXE;yS)CyddYSiEnX^Mim%1 zF`YXTuf;eZM@FS(F|82HU*&`tx#Z1;phWBYet$Mvo{b%OaZV&FuTgRi=Sz)|^6aB= zu0RqPh@{mfPf_# z`D(Y^k7N(Ej^Ho7BCOjuy$70u2vSCD*Uf2ng#gJ1_loow(t)kp<5JhJO@PpLC4wlpHkEVnqk&N1E;su}P#3X{_jYi@ck&Wq zM-B108=>{w#>~T`-886pDWVp%JL76z{!0~nwfT39)c-a0#)p^AY|3 zn|zNWV`}ru$()FTnd9GY4-oys@>)B6)6R@y*5BN=h^euyiRnKiv7^&B1O6MvZ54l3 zQz2Rr3*!3DXKa=PJu#0emdt{bnp`V^*(`x0SRocC5$2>iw~k%Zlri_cF=s@M;#-OE z4=!d9wz`-xM1R5&H)pr|B=?qkcRh9Cd`fOYJbW#AJOGJYYK59YA4RZ35=2u{Ek@OZ z0zkn`7F9uECBXwg$*+}Mk|x;;BOZHfyo?}!9Ej5hbHH5y1(<+fGmbrlU#5#tgu|3C zPDmKZR)~#&5$bYHR7eXpfherXh()WU!y`V-yCJdw+b4S+GFo~ zi=AN(Lb>JVaal=>!B*t2yt3^CV}1%YL7o#lH;EQlf7a7#X7|b4(j(nEa%OQLpiR}H zgRuG)P~N+}>M#=w#^N 1$, dann folgt direkt + \begin{salign*} + \int_{\R^{n}}^{} |f|^{p} \d{x} &= \int_{\R^{n}}^{} |f|^{p-1} |f| \d{x} \\ + &\le \sup_{x \in \R^{n}} |f|^{p-1} \int_{\R^{n}}^{} |f| \d{x} \\ + &\stackrel{\text{\ref{le:2}}}{\le} \Big( \underbrace{\sup_{x \in \R^{n}} |f|}_{ < \infty} \Big)^{p-1} + \underbrace{\Vert f \Vert_{L^{1}}}_{< \infty} \\ + &< \infty + .\end{salign*} + Setze nun $S \coloneqq \sup_{x \in \R^{n}} \left| \prod_{i=1}^{n} (1+x_i^2) f \right| $. + Es ist $\prod_{i=1}^{n} (1+x_i^2) = \sum_{\alpha \le 1} x^{\alpha}$. Dabei bezeichne + $\alpha \le 1 \iff \alpha_i \le 1$ $\forall i \in \{1, \ldots, n\} $. Damit folgt + \begin{salign*} + S &= \sup_{x \in \R^{n}} \left| \prod_{i=1}^{n} (1+x_i^2) f \right| \\ + &= \sup_{x \in \R^{n}} \left| \sum_{\alpha \le 1} x^{\alpha} f \right| \\ + &\le \sum_{\alpha \le 1} \underbrace{\sup_{x \in \R^{n}} |x^{\alpha} f |}_{< \infty} \\ + &< \infty \tageq \label{e1} + .\end{salign*} + Es ist außerdem $\prod_{i=1}^{n} (1+x_i^2) \ge 0$ und messbar, also Fubini anwendbar. Damit + folgt nun + \begin{salign*} + \int_{\R^{n}}^{} |f| \d{x} &= \int_{\R^{n}}^{} \prod_{i=1}^{n} (1 + x_i^2)|f| \frac{1}{\prod_{i=1}^{n} (1+x_i^2) } \d{x} \\ + &\le S \int_{\R^{n}}^{} \frac{1}{\prod_{i=1}^{n} (1+x_i^2) }\d{x} \\ + &\stackrel{\text{Fubini}}{=} S \prod_{i=1}^{n} \int_{\R}^{} \frac{1}{1+x_i^2} \d{x_i} \\ + &\stackrel{\text{\ref{le:1}}}{=} S \pi^{n} \\ + &\stackrel{\text{(\ref{e1})}}{<} \infty + .\end{salign*} + Also $f \in L^{1}(\R^{n})$. + \item Seien $f, g \in \mathscr{S}(\R^{n})$. Dann sind $f, g \in C^{\infty}(\R^{n}, \mathbb{C})$ und + mit Produktregel auch $fg \in C^{\infty}(\R^{n}, \mathbb{C})$. Seien nun + $\alpha, \beta \in \N_0^{n}$. Dann ex. nach Produktregel ein + $N \in \N$ und $\lambda_i \in \N_0$ mit $\nu_i, \mu_i \in \N_0^{n}$, s.d. + \begin{salign*} + |\partial_{\beta} (fg) | &= \sum_{k=1}^{N} \lambda_i (\partial_{\nu_i} f)(\partial_{\mu_i} g) + .\end{salign*} + Damit folgt nun + \begin{salign*} + \sup_{x \in \R^{n}} |x^{\alpha} \partial_{\beta} (fg)| + &= \sup_{x \in \R^{n}} \left|x^{\alpha} \sum_{i=1}^{N} \lambda_i (\partial_{\nu_i} f)\left( \partial_{\mu_i }g \right) \right| \\ + &\le \sum_{i=1}^{N} \lambda_{i} \sup_{x \in \R^{n}} |x^{\alpha} (\partial_{\nu_i} f)(\partial_{\mu_i}g)| \\ + &\stackrel{\text{\ref{le:2}}}{\le} + \sum_{i=1}^{N} \lambda_i \underbrace{\sup_{x \in \R^{n}} |x^{\alpha} \partial_{\nu_i} f|}_{< \infty} + \cdot \underbrace{\sup_{x \in \R^{n}} | \partial_{\mu_i} g |}_{< \infty} \\ + &< \infty + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\begin{lemma} + Sei $f \in \mathscr{S}(\R)$. Dann gilt für $\xi \in \R$ + \[ + \widehat{\frac{\mathrm{d} f}{\d{x}}}(\xi) = i \xi \widehat{f} + .\] + \label{le:fourier} +\end{lemma} +\begin{proof} + Zunächst sei $\xi \in \R$. Dann betrachte + \[ + g_n \coloneqq \frac{\partial}{\partial x} f e^{- i \xi x} \chi_{[-n, n]} + .\] Dann ist + \begin{salign*} + |g_n| &= \left| \frac{\partial}{\partial x} f e^{-i\xi x} \right| \chi_{[-n, n]} \\ + &\le \left| \frac{\partial}{\partial x} f e^{-i\xi x} \right| \\ + &= \left|\frac{\partial f}{\partial x} e^{-i \xi x} - i k f e^{-i \xi x} \right| \\ + &\le \left| \frac{\partial f}{\partial x} e^{-i \xi x}\right| + + \left| i k f e^{-i \xi x} \right| \\ + &\stackrel{|e^{-i \xi x}| = 1}{=} \left| \frac{\partial f}{\partial x} \right| + |i\xi| |f| + .\end{salign*} + Die rechte Seite ist in $L^{1}(\R)$, denn $f$, $\partial_x f \in \mathscr{S}(\R)$ und + damit nach Aufgabe 1 $f, \partial_x f \in L^{1}(\R)$ und damit insbesondere + $\left|\frac{\partial f}{\partial x}\right| + |i \xi| |f| \in L^{1}(\R)$. + Da außerdem $g_n \to \frac{\partial}{\partial x} f e^{- i \xi x}$ und $g_n$ messbar, + folgt mit dem Satz von der + dominierten Konvergenz und dem Hauptsatz ($g_n$ auf kompaktem Intervall $[-n, n]$ stetig und + R.-integrierbar, d.h. Lebesgue und R-Integral stimmen überein): + \begin{salign*} + \left| \int_{\R}^{} \frac{\partial}{\partial x} f e^{-i \xi x} \d{x}\right| + &= \left|\lim_{n \to \infty} \int_{\R}^{} g_n \d{x} \right| \\ + &= \left| \lim_{n \to \infty} \int_{-n}^{n} \frac{\partial}{\partial x} f e^{-i \xi x} \d{x} \right| \\ + &= \left| \lim_{n \to \infty} \left[f(n) e^{-i \xi n} - f(-n) e^{i \xi n} \right] \right| \\ + &\stackrel{|e^{-i \varphi}| = 1}{\le } \lim_{n \to \infty} |f(n)| + \lim_{n \to \infty} |f(-n)| \\ + &\stackrel{f \in \mathscr{S}(\R)}{=} 0 \tageq \label{e:3} + .\end{salign*} + Damit folgt nun + \begin{salign*} + \widehat{f'} &= \frac{1}{\sqrt{2\pi} } \int_{\R}^{} f'(x) e^{-i \xi x} \d{x} \\ + &= \frac{1}{\sqrt{2\pi} } \int_{\R}^{} \frac{\partial}{\partial x} f(x) e^{-i \xi x} \d{x} + - \frac{1}{\sqrt{2\pi} } \int_{\R}^{} f \frac{\partial }{\partial x} e^{-i \xi x} \d{x} \\ + &\stackrel{\text{(\ref{e:3})}}{=} i \xi \widehat{f} + .\end{salign*} +\end{proof} + +\begin{aufgabe} + Sei zunächst $j \in \N_0$ und $x \in \R$ beliebig. Dann ist $H_1(x) = 2x = 2x H_0(x) - H_0'(x)$. + Für $j > 0$ gilt + \begin{salign*} + H_j'(x) &= (-1)^{j} 2x e^{x^2} \frac{\mathrm{d}^{j}}{\d{x^{j}}} e^{- x^2} + + (-1)^{j} e^{x^2} \frac{\mathrm{d}^{j+1}}{\d{x^{j+1}}} e^{-x^2} \\ + &= 2x H_j - H_{j+1} \tageq \label{e2} + .\end{salign*} + Das zeigt die im Hinweis behauptete Identität. Damit folgt nun für $j \in \N_0$ + \begin{salign*} + \psi_j' &= H_j' e^{- \frac{x^2}{2}} - x H_j e^{- \frac{x^2}{2}} \\ + &= H_j' e^{- \frac{x^2}{2}} - x \psi_j \\ + &\stackrel{\text{(\ref{e2})}}{=} + (2x H_j - H_{j+1}) e^{-\frac{x^2}{2}} - x \psi_j \\ + &= 2x \psi_j - \psi_{j+1} - x \psi_j \\ + &= x \psi_j - \psi_{j+1} \tageq \label{e:4} + .\end{salign*} + Das zeigt die linke Identität. Es gilt weiter + \begin{salign*} + \left( \widehat{\psi_j} \right) '(\xi) + &= \frac{\mathrm{d}}{\d{\xi}} \frac{1}{\sqrt{2 \pi} } \int_{\R}^{} \psi_j(x) e^{-i \xi x}\d{x} \\ + &= \frac{1}{\sqrt{2\pi} } \int_{\R}^{} \psi_j \frac{\partial}{\partial \xi} e^{-i \xi x}\d{x} \\ + &= - i \frac{1}{\sqrt{2\pi} } \int_{\R}^{} x \psi_j e^{-i \xi x} \d{x} \\ + &= - i \widehat{x \psi_j} \\ + &\stackrel{\text{(\ref{e:4})}}{=} - i \widehat{\psi_{j+1}} -i \widehat{\psi_{j}'} \\ + &\stackrel{\text{\ref{le:fourier}}}{=} - i \widehat{\psi_{j+1}} -i (i \xi \widehat{\psi_j}) + \intertext{Damit folgt} + \widehat{\psi_{j+1}} &= i (\widehat{\psi_j})' - i \xi \widehat{\psi_j} \\ + &= - i ( \xi \widehat{\psi_j} - (\widehat{\psi_j})') \tageq \label{e:5} + .\end{salign*} + Folgere die Behauptung nun per Induktion nach $j$. + Für $j=0$ ist $\psi_0 = H_0 e^{- \frac{x^2}{2}} = e^{- \frac{x^2}{2}}$. Also $\psi_0$ ist + die Gaußfunktion und damit folgt $\widehat{\psi_0} = \lambda_0 \psi_0$ mit $\lambda_0 := 1$. + Sei die Behauptung nun gezeigt für $j \in \N$. Dann gilt ausgehend von (\ref{e:5}) für + $\xi \in \R$: + \begin{salign*} + \widehat{\psi_{j+1}}(\xi) &= - i(\xi \widehat{\psi_j}(\xi) - (\widehat{\psi_j})'(\xi)) \\ + &\stackrel{\text{IV}}{=} -i (\xi \lambda_j \psi_j(\xi) - \lambda_j \psi_j'(\xi)) \\ + &= -i \lambda_j ( \xi \psi_{j}(\xi) - \psi_j'(\xi)) \\ + &\stackrel{\text{(\ref{e:4})}}{=} -i \lambda_j \psi_{j+1} + .\end{salign*} + Mit $\lambda_{j+1} \coloneqq - i \lambda_j \in \{\pm 1, \pm i\}$ folgt die Behauptung. +\end{aufgabe} + +\begin{aufgabe} + Seien $\alpha > 0$ und $y \in \R^{n}$. + Zunächst ist wegen $|e^{-i \varphi}| = 1$ für $\varphi \in \R$ und + $f, \tau_y f(x), \delta_{\alpha} f(x) \in \mathscr{S}(\R^{n})$ und + Aufgabe 1 (b) auch + $f e^{-i \xi x}$, $\tau_y f e^{-i \xi x}$, $\delta_{\alpha} f e^{-i \xi x} \in L^{1}(\R^{n})$. + Sei im folgenden $\xi \in \R^{n}$. + \begin{enumerate}[a)] + \item Mit der Vorbemerkung ist der Transformationssatz anwendbar mit $z = x - y$. Dann folgt: + \begin{salign*} + \widehat{\tau_y f}(\xi) &= \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} f(x-y) e^{-i \xi \cdot x} \d{x} \\ + &\stackrel{\text{Trafo}}{=} \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} f(z) e^{-i \xi \cdot (z + y)} \d{z} \\ + &= e^{-i \xi \cdot y} \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} f(z) e^{-i \xi \cdot z} \d{z} \\ + &= e^{-i \xi \cdot y} \widehat{f}(\xi) + .\end{salign*} + \item Mit der Vorbemerkung ist der Transformationssatz anwendbar mit $z = \alpha x$. Dann + ist $\text{det}(Dz) = \alpha ^{n}$ und es folgt + \begin{salign*} + \widehat{\delta_\alpha f}(\xi) &= \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} f(\alpha x) e^{-i \xi x} \d{x} \\ + &\stackrel{\text{Trafo}}{=} \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} \frac{1}{\alpha ^{n}} f(z) e^{- i \xi \cdot \frac{z}{\alpha}} \d{z} \\ + &= \frac{1}{\alpha ^{n}} \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} f(z) e^{- i \left( \frac{\xi}{\alpha} \right)\cdot z } \d{z} \\ + &= \frac{1}{\alpha ^{n}} \widehat{f}\left( \frac{\xi}{\alpha} \right) + .\end{salign*} + \item Es ist $f, g \in L^{1}(\R^{n})$. Damit folgt + \begin{salign*} + \int_{\R^{n}}^{} \int_{\R^{n}}^{} |f(y) g(y-x) e^{-i \xi \cdot x}| \d{y} \d{x} + &= \int_{\R^{n}}^{} \int_{\R^{n}}^{} |f(y)g(y-x)| \d{y} \d{x} \\ + &= \int_{\R^{n}}^{} (|f| * |g|)(x) \d{x} \\ + &= \Vert |f| * |g| \Vert_{L^{1}(\R^{n})} \\ + &\stackrel{\text{Zettel 7}}{<} \infty + .\end{salign*} + Damit ist Fubini anwendbar und es gilt + \begin{salign*} + \widehat{f \cdot g}(\xi) &= \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} \left[ \int_{\R^{n}}^{} f(y) g(x-y) \d{y} \right] e^{- i \xi \cdot x} \d{x} \\ + &\stackrel{\text{Fubini}}{=} + \frac{1}{(2\pi)^{\frac{n}{2}}} \int_{\R^{n}}^{} \left[ \int_{\R^{n}}^{} f(y)g(x-y) + e^{-i \xi \cdot x}\d{x} \right] \d{y} \\ + &= \int_{\R^{n}}^{} f(y) \left[ \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R^{n}}^{} \tau_y g(x-y) e^{-i \xi \cdot x} \d{x} \right] \d{y} \\ + &= \int_{\R^{n}}^{} f(y) \widehat{\tau_y g}(\xi)\d{y} \\ + &\stackrel{\text{(a)}}{=} \widehat{g}(\xi) \int_{\R^{n}}^{} f(y) e^{- i \xi \cdot y} \d{y} \\ + &= (2\pi)^{\frac{n}{2}} \widehat{g}(\xi) \widehat{f}(\xi) + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\begin{lemma} + Seien $n, m \in \N$ mit $m \le n$ und $g \in \mathscr{S}(\R^{m})$. Dann ist + $g \circ \tau_{n}^{m} \in \mathscr{S}(\R^{n})$, wobei $\tau_{n}^{m}\colon \R^{n} \to \R^{m}$ mit + $\tau_n^{m}(x_1, \ldots, x_n) = (x_1, \ldots, x_m)$ für $x \in \R^{n}$. + + Für $m = 1$ ist weiterhin $g \circ \pi_i^{n} \in \mathscr{S}(\R^{n})$, wobei + $\pi_i^{n}\colon \R \to \R^{n}$ mit $\pi_i^{n}(x_1, \ldots, x_n) = x_i$ für $x \in \R^{n}$ + und $i \in \{1, \ldots, n\} $. + \label{le:tau} +\end{lemma} +\begin{proof} + Seien $\alpha, \beta \in \N_0^{n}$ beliebig. Falls $\beta_j \neq 0$ für ein $m < j \le n$. Dann + ist $\partial{_\beta} \tau_i = 0$ $\forall i = 1, \ldots, m$. Also + auch $\partial_{\beta} (g \circ \tau) = 0$. Sei also $\beta_j = 0$ für $m < j \le n$ + und bezeichne $\nu \coloneqq (\beta_1, \ldots, \beta_m)$. Weiter + gilt für $x \in \R^{n}$, dass $|x^{\alpha}| = |x_1^{\alpha_1} \cdot \ldots \cdot x_n^{\alpha_n}| \le + |x_1^{\alpha_1} \cdot \ldots \cdot x_{m-1}^{\alpha_{m-1}} \cdot \Vert x \Vert_{\infty}^{\sum_{i=m}^{n} \alpha_i}|$. Setze + also $\mu \coloneqq (\alpha_1, \ldots, \alpha_{m-1}, \sum_{i=m}^{n} \alpha_i)$ + Dann + folgt + \begin{salign*} + \sup_{x \in \R^{n}} | x^{\alpha} \partial_{\beta} (f \circ \tau) | + \le \sup_{x \in \R^{m}} | x^{\mu} \partial_\nu f | < \infty + .\end{salign*} + Im Fall $m = 1$ existieren die $\pi_i^{n}$ und die Argumentation verläuft exakt analog. +\end{proof} + +\begin{aufgabe} + Zeige zunächst induktiv, dass $f \in \mathscr{S}(\R^{n})$. + Für $n = 1$ trivial, denn $f = f_1 \in \mathscr{S}(\R)$. Sei nun $n \in \N$ beliebig und + Aussage gezeigt. Dann betrachte $\tau_{n}^{n+1}$, $\pi_i^{n+1}$ und $\pi_i^{n}$ + aus \ref{le:tau}. + \begin{salign*} + f &= \prod_{i=1}^{n+1} (f_i \circ \pi_i^{n+1}) \\ + &= \prod_{i=1}^{n} (f_i \circ \pi_i^{n+1}) \cdot (f_{n+1} \circ \pi_{n+1}^{n+1}) \\ + &= \overbrace{\tau_n^{n+1} \Big[ \underbrace{\prod_{i=1}^{n} (f_i \circ \pi_i^{n})}_{\in \mathscr{S}(\R^{n}) \text{ nach IV}} \Big]}^{\in \mathscr{S}(\R^{n+1}) \text{ nach \ref{le:tau}}} \cdot \underbrace{(f_{n+1} \circ \pi_{n+1}^{n+1})}_{\in \mathscr{S}(\R^{n+1}) \text{ nach \ref{le:tau}}} + .\end{salign*} + Nach Aufgabe 1 (c) folgt damit, dass $f \in \mathscr{S}(\R^{n+1})$. Das beendet die Induktion. + + Nun ist insbesondere $f \in L^{1}(\R^{n})$, d.h. Fubini ist anwendbar. Damit folgt für $x \in \R^{n}$: + \begin{salign*} + \widehat{f}(\xi) &= \frac{1}{(2\pi)^{\frac{n}{2}}} \int_{\R^{n}}^{} \prod_{i=1}^{n} f_i(x_i) + e^{- i \xi \cdot x} \d{x} \\ + &\stackrel{\text{Fubini}}{=} \frac{1}{(2\pi)^{\frac{n}{2}}} + \int_{\R}^{} f_1(x_1) \d{x_1} \int_{\R}^{} \ldots \int_{\R}^{} f_n(x_n) e^{- i \xi \cdot x} \d{x_n} \\ + &\stackrel{\xi \cdot x = \sum_{i=1}^{n} \xi_i x_i}{=} + \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi} } \int_{\R}^{} f_i(x_i) e^{- i \xi_i x_i} \d{x_i} \\ + &= \prod_{i=1}^{n} \widehat{f_i}(\xi_i) + .\end{salign*} +\end{aufgabe} + +\end{document}