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\underline{Fourier-Reihe} von $f$ ist $$\sum_{k = -\infty}^{\infty}c_k e^{ikx}$$ mit der n-ten Partialsumme $$s_n(x) = s_n(f,x) \coloneqq \sum_{k = -n}^{n}c_ke^{ikx}.$$ Die Fourier-Reihe läßt sich in der Form schreiben $$\frac{a_0}{2} + \sum_{k = 1}^{\infty}(a_k\cos(kx) + b_k \sin(kx))$$ wobei \begin{align*} a_k &= \frac{1}{\pi} \int_0^{2\pi} f(x) \cos(kx) \d x\\ - b_k &= \frac{1}{\pi} \int_0^{1\pi} f(x) \sin(kx)\d x + b_k &= \frac{1}{\pi} \int_0^{2\pi} f(x) \sin(kx)\d x \end{align*} \end{definition} \begin{satz} @@ -238,7 +238,7 @@ $$F_n(x) = \int_\pi^x \frac{1}{2 \sin\left(\frac{y}{2}\right)} \cdot \sin\left(\ \end{proof} \begin{satz} Sei $f\in R[0,2\pi]$ eine $2\pi$-periodische Treppenfunktion. Dann konvergiert die Fourier-Reihe von $f$ im quadratischen Mittel gegen $f$ und es gilt die Parsevalsche Gleichung (sog. Vollständigkeitsrelation) - $$2 \pi \underbrace{\int_0^{2\pi} |f(x)|^2\d x}_{=\qnorm{f}} = \sum_{k = -\infty}^{\infty}|c_k|^2$$ + $$\frac{1}{2\pi} \underbrace{\int_0^{2\pi} |f(x)|^2\d x}_{=\qnorm{f}} = \sum_{k = -\infty}^{\infty}|c_k|^2$$ \end{satz} \begin{proof} O.B.d.A. sei $f$ reellwertig (sonst werden Real- und Imaginärteil getrennt behandelt) und $|f(x)| \leq 1 \forall x\in [0, 2\pi]$ (sonst betrachte $\overline{f}(x) \coloneqq \frac{f(x)}{M},\; M = \sup\limits_{x\in [0, 2\pi]} |f(x)|$). diff --git a/analysisII.pdf b/analysisII.pdf index 15a186ebde5bb2a625ffd5f90298f6a25aa78d7e..737d9e4eedfa15a220e1835b43f1c6ba0af7056e 100644 GIT binary patch delta 7225 zcmai&MNpg#7o~CcAdR~d+})vZcX#&?{6!ikKyW9xy99zma7%)_Hm)JK%~$nT%`9i} ztj@DMi*xVRF6qTCX=p2Kg##>Krc@j(6L4;1>b=&E7kuBaUBdervQ26sk}dK*wL^7I z*l~BW(>qm^iLkCQV_zpj`+Hw6G6EQuMi$(Dax7zRB?loyKU>*o=?e0CJK&ov7hvSg zwc~xh+t0m8Wvo!K6&mMTT6we`sianEnqpF}Jy?093ly*jc-VP)W~?xRu$EC}0O(^? zi`_4?r$mRd04NcH11}#%TQ|M!Gj_hcfW0LyakKRvbT<{893e1k9<&tnxaIUa8}N9# zS(&^cYM8mgF>a3&7it?=q-7v3QnIIYyDGaP5xrI7Ca#@_7_C)46tL(u#3m-lRU6)@ zUONUo`+w@n3V^b2q0?sGHW*w1tV5_JwT~-uJ>gcP>nNvsiZr0;df9_N4L_95*uhv_ z12nq=;v4N;`m%~EE~08KwJ1x%om^|nS<$3`7#Y{{nYDB=eNOXbUxW_zi+ao1`M7%9o~&J!+_4F+txJU9Zz}d5 zjJfd%B9FInuUj1411mMq4jLNLpqrnVs6o5VK>}To;qo0eV?~hcr*MZs1{F)%2o8sm zXWRQ~f^pq#6=tCr)N(}N&{PRB11*=?HFomNR+142wZqo746|*jk|rPle=YM{Qi(EH zr}AEll{Z&PEhx?FGgPd*`zAucvQF;#L%#;9ss_c4CS&|dT4U|uCq zcpr?3{b003DOIoMp7ts(MbnD%mMh7((;E}yhkT7eg{T30G?hBsW1gO&k$i%m$4Lb1 zX%FA=dc}$p8xsYwwwBMT17*gjTYIi5VmUFVncX6}&KT~#pGzgo$(xu^iz7L+*Soj7 zjA86J_tR3NW=y3fG-6c>s>>h7L0e3zwaJ5Ga=n_)aYzWldq=5%Y~o6z*0Wf?(Mo}F z8E)X4(lh`iebkitR5oRAmB||vkLU%}+6K3iLzCO$zwEjl{oLhoGqgpx0Vi#ldn;!n zqqAVc`A^Q8AuNQS8uJODku|0g(_1ApGBJG6c@EMovvLFSxJCZ>US+9~$eMPO*B`8K zzT4;Cy{wuIr(vLh-sGx`AjS2l`ax-QW3sD&oj2fh7f-1vFJrjbK<5!no;#+bQAT&- zn#4g@&nU5CSib|)33=;_{@W18aH%3w8ZBRCV4joq>1~sI;f@61M8?uWG}wRdqpqJE z3#L1fe$C&2oxW?r^3FgyQzg}3=3#8%Vi;e`-4eRhEb0&JETQd{?D54{h&CN^4e_Q{PD0Q=x`$ z@0So~Qq(2>-1fCIS!lFRaY(mWx_Oim_6WeIRAf562Y2zg3!+9#nPG*s^6~}rwWOr- zhoGAXviUK6#~|O5M^x4>KVY4W29^acV*$@B-kEUKHbm&Q50fd<@TwaT@}YC8I~N&p zzMa}aqkZ{g5vlhq6}T}53Z3;D`iu_DiJo6#Z4G^6=1o}Un2F&+ISWuDl>9LaKjFep-;3aTEU6%|u{tNe!&u}%xqO3;QaX$aWb zqY#On$)3;6bdn|FyW0WGZ@@(qt`S4L!x}kO0?i(SA^Wn>`cULYg zSFR1Q=rs2qUY0lucJJq?s+eaQYAusxqApWqogjP4C1EAj(SL1TCc`CfYW1V-bO^Yh zy1gPJ8aL%72tB5~6ObzP-LI~K}-38hL#>*6AT_XX>dQ4lCXMX_BY?xU2V>o=4g!KpJx6e7a-ZD(E{13 z5|!eIU`EXsPX^6s%H3ozzJvu{`aeUp*3$|!C8w@;z27CPB|yq6u_a1+eQnf%Q$uF& zK7~7Y?4PoE@B;lZC`Di?ZGTYwD8E5IP5Q{F&-&(jz@hTR?M$Zs{_noYx3&Dl8xO3- zB>jal(NrIPD|rQ_AC53F&3w8Q3If)Bfg<}{oXm~O6D1aSnOQY2lqwXE@pCSGA@-x! zH{GnWDdeYeL12nWT52G06x#$3aqdjyr=>p@>{EwKp5$+j82E)O+%9{;=?>`5;fA=Y z6o~52yxh;V>l5sG=8R6}_+xSvCi2BMV}=h%I)DDDGK7+pJu7@b!FVc4MeRDYv8q!# zd4hJ+sAfxQfR_myBj9W;(NjUpIiq3hbF1WOLJ? 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