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Lecture 26


¶Ç²ÎªºÀ³¥Î¼Æ¾Ç, «üªº¬O±N¼Æ¾ÇÀ³¥Î¦bª«²z¤è­±. ¨ä¹ê³o¤]¬O¤û¹y, ¼Ú©Ô³o¨Ç¦­´Áªº¤Hª«©Ò±Mª`ªº°ÝÃD. ³o¤@Ãþ°ÝÃD¾É¤Þ¥X¨Óªº¼Æ¾Ç°ÝÃD¤j­P¤W¬O·L¤À¤èµªº¨D¸Ñ°ÝÃD. °ª´µ¾É¥Xªº³Ì¤p¥­¤è®t¤èªk, ¦b¤µ¤Ñ¥i¥H¥Î¨Ó³B²z²Î­p©M¸gÀپǤW¤è­±ªº°ÝÃD. ¦ý¬O·íªì¥Lµo©ú³o®M¤èªkªº¥Øªº¬O¬°¤F±À´ú¤@­Ó¤Ñ¤å¾Ç¤Wªº²¶H. °ò¥»¤W¥i¥H»¡, ¤G¤Q¥@¬ö«eªºÀ³¥Î¼Æ¾Ç³£¦b³B²z¦ÛµM¬ì¾Çªº°ÝÃD. ¦b¤G¤Q¥@¬öªº¤¤¬q, ¤j­P¤W´N¬O¤G¦¸¤j¾Ô¤§«á, ²Î­p¾Ç»P­pºâ¾÷¬ì¾Çªº§Ö³tµo®i, ±a°Ê¤F¹³¼Æ¾Ç³W¹º, Â÷´²¼Æ¾Ç»P¹Ï½×³o¨Ç¼Æ¾Ç¤À¤äªºµo®i. ¶i¦Ó¤]±N¼Æ¾ÇªºÄ²¨¤¦ù¤J¤FªÀ·|¬ì¾Çªº»â°ì. ¦Ó¹q¤l­pºâ¾÷ªºÀ³¥Î, ¦ÛµM²£¥Í¤FÂ÷´²ªº, ¼Æ¦ì¤Æªº·s¸ê®Æ§ÎºA. ¥i¥H»¡, ¼Æ¦ì¤ÆªºÁn­µ»P¼v¹³, ¬O³Ìªñ¤T¤Q¦ ¤ µo¥Íªº. °w¹ï³o¨Ç·sªº¸ê®Æ§ÎºA²£¥Í¤F·sªº°ÝÃD, »Ý­n·sªº¤èªk. ³o¤@ªù·sªº¾Ç°Ý, ²ÎºÙ¬°°T¸¹³B²z (signal processing). ¦Ó³oªù¬ì¾Çªº®Ö¤ßª«¥ó¤§¤@, ´N¬O³Å¥ß¸­1¨Ú¦P´Áªºªk°ê¤H, ´¿¸g¬O®³¯¨Úªº±s¥Î¬ì¾Ç®a, ÀH­x»·©º®J¤Î, ¨Ã¹ï¥j®J¤Î¤å¤Æªº¬ã¨s¦³©Ò°^Äm. ¥L©Òµo±¸ªº¤@¥óµÛ¦W«´§Î¤å¦rªdª©, Rosetta stone, ¦b¥L³Q­^°ê®ü­x«R¸¸ªº®É­Ôµ¹¨S¦¬¤F, ²¦b®i¥Ü©ó¤j­^³Õª«À]. ¥LªºÃ­©wªº¬ì¾Ç®a¥Í¬¡©l©ó®³¯¨Ú³Q¬y©ñ«n¤j¦è¬vªº¤p®q (1814). ¦ý¬O¥L¦b 1807 ´N¤w¸g´£¨ì¹L³o¤@Ãþªº¯Å¼Æ. ¯Å¼Æ. ¤]´N¬O»¡, À³¥Î¼Æ¾Çªº»â°ì, ±q¤T¦Ê¦ «eªºª«²z¤O¾Ç, º¥º¥¦aÂX¤j, ¨ì¤µ¤ÑÁÙ¥]§t¤F°T¸¹³B²z.

©Ò¿×³Å¥ß¸­¯Å¼Æ (Fourier series) ¬O«ü

\begin{displaymath}a_0 + \sum_{n=1}^\infty a_n \cos nx + b_n \sin nx \eqno(1)
\end{displaymath}

³o¼Ëªº¯Å¼Æ. ¦pªG¥u¨ú³¡¤À©M, «hºÙ¬°¤T¨¤¦h¶µ¦¡ (trigonometric polynomial). ¨ä¤¤ ak, bk ³£¬O¹ê¼Æ, ºÙ¬°³Å¥ß¸­«Y¼Æ (Fourier coefficients). ¥¦­º¦¸¥X²©ó¼Ú©Ô (Euler) ªº¤@­Óµ¥¦¡

\begin{displaymath}{x\over 2} = \sin x - {1\over2} \sin 2x + {1\over3}\sin 3x - \cdots,
\quad x\in (-\pi, \pi).
\end{displaymath}

¦ý¬O¼Æ¾Ç¥v¤W¨Ã¨S¦³¥H¼Ú©Ô¨Ó©R¦W³o¤@Ãþªº¯Å¼Æ. ³o©Î³O¦]¬°³Å¥ß¸­µo²³oÃþ¯Å¼Æªº­ì¦]¨ã¦³¤ñ¸û²`ªº¼Æ¾Ç¼vÅT.

¥Ñ©Ò¿×ªº¼Ú©Ô¤½¦¡

\begin{displaymath}e^{i\theta} = \cos\theta + i\sin\theta,
\end{displaymath}

§Ú­Ì¥i¥H§ï¼g

\begin{displaymath}\cos nx = {e^{inx} + e^{-inx}\over 2},\quad
\sin nx = {e^{inx} - e^{-inx}\over 2}.
\end{displaymath}

©Ò¥H, ³Å¥ß¸­¯Å¼Æ (1) ¤S¥i¥H¼g¦¨

\begin{displaymath}\sum_{n=-\infty}^\infty c_n e^{inx}.
\end{displaymath}

³o®É x ¬O¹ê¼Æ, ¦Ó cn ¬O½Æ¼Æ.

®õ°Ç¯Å¼Æ©M³Å¥ß¸­¯Å¼Æ³£¬O§â¤@­Ó¨ç¼Æ©î¦¨µL½a¦h¶µªº¤èªk. ³o¨ÇµL½a¦h¶µªº§C¦¸¶µÁ`¬O¤ñ¸û­«­n¦Ó°ª¦¸¶µ¶V¨Ó¶V¤p, ¥i¯à¬Ù²¤¤F¤]¤£·|³y¦¨¤Ó¤jªº» ®t. ®õ°Ç¯Å¼Æ§â¨ç¼Æ¤À¸Ñ¦¨¥H¬Y­ÓÂI¬°¤¤¤ßªº¦h¶µ¦¡, ³Å¥ß¸­¯Å¼Æ§â¨ç¼Æ¤À¸Ñ¦¨¤£¦PªºÀW²vªi¬q. ®õ°Ç¯Å¼Æ³q±`¥u¦b¬Y­ÓÂIªºªþªñ¹Gªñ¨ç¼Æ, ³Å¥ß¸­¯Å¼Æ³q±`¥u¦b¬Y­Ó°Ï¶¡¤º¥­§¡¦a¹Gªñ¨ç¼Æ. ®õ°Ç¯Å¼Æªº·¥­­¨ç¼Æ (¦pªG¦s¦bªº¸Ü), ¤@©wµ¥©ó­ì¨ç¼Æ, ³Å¥ß¸­¯Å¼Æªº·¥­­¨ç¼Æ, ¤£¨£±o¦b¨C­ÓÂI³£µ¥©ó­ì¨ç¼Æ.

½Ò¥» 10.6 ³Å¥ß¸­¯Å¼Æ


±q (1) ¦¡¥i¥H¬Ý¥X, ³Å¥ß¸­¯Å¼Æ¥²©w¬O¤@­Ó $2\pi$ ©P´Á¨ç¼Æ. ³q±`§Ú­Ì¥u¬Ý¥¦¦b $[-\pi, \pi]$ ¤§¶¡ªº¹Ï§Î. ¦pªG§Ú­Ì©Ò­n³B²zªº¨ç¼Æ f(x) ¤£¬O­Ó $2\pi$ ©P´Á¨ç¼Æ, ³q±`§Ú­Ì¥ý¨M©w­n³B²z­þ­Ó°Ï¬q, ¤ñ¦p»¡ [a,b], µM«á±N¤§¥­²¾©ÔÁY¨ì $[-\pi, \pi]$, ¦A±N¥¦©P´Á¤Æ, µM«á¨ú¨ä³Å¥ß¸­«Y¼Æ. ¨D³Å¥ß¸­«Y¼Æªº¤èªk, ¦C¦b½Ò¥» 637 ­¶. ©Ò¿×ªº©P´Á¤Æ, ´N¬O§â¨ç¼Æ¦b $[-\pi, \pi]$ ªº¹Ï§Î, ¤À¤ù§Û¨ì $[-3\pi, -\pi]$, $[\pi, 3\pi]$, $\cdots$ ³o¨Ç°Ï¶¡¥h. ¤]´N¬Oµw§â¥¦Åܦ¨ $2\pi$ ©P´Á¨ç¼Æ. °£«D­ì¨ç¼Æ f(a) = f(b), §_«h³Q©P´Á¤Æ¤§«áªº¨ç¼Æ¬O¤£³sÄòªº.

¨Ò¦p, ¥O

\begin{displaymath}f(x) = \cases{
{1\over4} - \vert x\vert & if $\vert x\vert < {1\over4}$,
\cr
0 & otherwise}\quad x\in [-\pi, \pi],\eqno(2)
\end{displaymath}

µM«á©Ý®i f(x) ¦¨ $2\pi$ ¶g´Á¨ç¼Æ. ¥i¨£ f(x) ¬O¤@­Ó°¸¨ç¼Æ, ©Ò¥H bn ³£¬O¹s. ­pºâ $a_0 = {1\over 32\pi}$, a1 ¨ì a16 ªº­È¤À§O¦p¤U

\begin{eqnarray*}&0.0099\quad0.0097\quad0.0095\quad0.0091\quad0.0087\quad0.0082
...
...ad0.0051\quad0.0044\quad0.0038\quad0.0031
\quad0.0026\quad0.0021
\end{eqnarray*}


¦¹³¡¤À©Mªº¹Ï§ÎÅã¥Ü¦b¹Ï¤­¤Q¤­.

\begin{displaymath}\vbox{\hsize 4truein\epsfxsize 4truein\epsfbox{samp.eps}
\centerline{\rm Fig~55} }
\end{displaymath}

¤]½Ð°Ñ·Ó½Ò¥»¤W½d¨Ò 1 »P½d¨Ò 2 ªº¹Ï§Î. ª`·N¦b¤£³sÄòÂIªº®ÇÃä, ³Å¥ß¸­¦h¶µ¦¡ªº¹Ï§Î·|¦³­Ó¦ü¥G©T©w°ª«×ªº¬ð°_.

·í n ¤pªº®É­Ô, §Ú­ÌºÙ an ©M bn ¬°§CÀW«Y¼Æ; ·í n ¤jªº®É­Ô, §Ú­ÌºÙ an ©M bn ¬°°ªÀW«Y¼Æ. ±q¤W­±ªº¨Ò¤l¸Ì, §Ú­Ì¬Ý¨ì, §CÀW«Y¼Æ (ªºµ´¹ï­È) ¤ñ¸û¤j, ¦Ó°ªÀW«Y¼Æ¤ñ¸û¤p. ¨ä¹ê³o¬O­Ó¤@¯ë©Êªº²¶H: ¥ô¦ó¨ç¼Æ (³q±`¦Ò¼°µ¤@µ§°T¸¹), ¨ä§CÀW³¡¤À¬O¤ñ¸û­«­nªº, ¦Ó°ªÀW³¡¤Àªº®¶´T«Ü¤p.2#53#> (1) ¦Ó¥B, ©Ò¿×ªºÂø°T³q±`¥X²¦b°ªÀW³¡¤À. (2) §Ú­Ì¤]¥i¥H³o¼Ë²z¸Ñ: ³y¦¨°ªÀW©Ò»Ý­nªº¯à¶q¤ñ¸û¤j, ¦pªG¤@¦@¥u¦³¦³­­¦hªº¯à¶q, ·íµM¯à¶q·|¶°¤¤¦b§CÀWªº³¡¤À. ³o­Ó``°ªÀW«Y¼Æ«Ü¤p''ªº²¶H, ¥i¥H±q¹Ï§Î¤WÆ[¹î. °²·Q¤@­Ó $2\pi$ ©P´Á¨ç¼Æ f(x), ¦b $[-\pi, \pi]$ ¤¤, ¨ä¹Ï§Î¦p¤U: ¹Ï¤­¤Q¤C©M¤­¤Q¤K¤À§O¬O $f(x) \sin x$ ©M $f(x) \sin 20x$ ªº¹Ï§Î. $\int_{-\pi}^\pi f(x) \sin x \,dx$ ¬O¥ª¹Ï¤¤ y ¶b¤W¤èªº­±¿n´î¥h¤U¤èªº­±¿n, ¨ä©w¿n¤À­È¤j¬ù¬O -2.5203. $\int_{-\pi}^\pi f(x) \sin 20x \,dx$ «h¬O¥k¹Ï¤¤ y ¶b¤W¤èªº­±¿n´î¥h¤U¤èªº­±¿n, ¥Ñ¹Ï¥i¨£¤W¤U³¡¥÷´X¥G¤¬¬Û®ø¥h, ©Ò¥HÀ³¸Ó«Ü¤p; ¨ä©w¿n¤À­È¤j¬ù¬O 0.0516.

¥H¤@±i¤H¹³ªñ·Ó¬°¨Ò, ¯à°÷±q¤­©x¥ §Î»¥X¨º­Ó¤H¬O½Ö, ÄÝ©ó§CÀW³¡¤À. ­n¼Æ¼Æ¥L²´¨¤¦³´Xµ·³½§À¯¾, ÀY¤W¦³´X²ô¥Õ¾v, ÄÝ©ó°ªÀW³¡¤À. ¥²¶·­n¤ñ¸û°ªªº¯à¶q (¤ñ¸û¶Qªº·Ó¬Û¾÷, ¤ñ¸û¶Qªº¨R¬ µ§Ç), ¤ ¯àÅã²³o¨Ç°ªÀW. ¦Ó§Y¨Ï¨S¦³³o¨Ç°ªÀW, ¤]¨Ã¤£¼vÅT§Ú­Ì¿ëÃѳo±i·Ó¤ù.

¤@­Ó $2\pi$ ©P´Á¨ç¼Æªº¯à¶q, ©M¥¦¤À§O¦b¦UÀW²vªi¬qªº¯à¶q, ©w¸q¦b½Ò¥»ªº 641 ­¶. ©Ò¿×ÀWÃФÀªR´N¬OÀ˵ø¤@­Ó¨ç¼Æ¦b¦UÀW²vªi¬qªº¯à¶q. §Ú­Ì¬Ý¨ì, °ªÀWªº¯à¶qÁ`¬Oº¥´î¨ì¹s, §_«h­ì¨ç¼Æ¥²¨ã¦³µL½aªº¯à¶q (¦ý¬O, §Ú­Ì»¬°¦ÛµM¬É¤¤À³¸Ó¨S¦³µL½a¯à¶qªº°T¸¹). ½Ò¥»¤¤ªíºt¤F¦p¦ó¥ÎÀWÃФÀªR¨Ó¤À¿ë½Ý²Ã©M¤p³â¥zªº­µ¦â.

³Å¥ß¸­¯Å¼Æ¦p¦ó¦¬ÀÄ, ¥i¥H»¡¬O¤Q¤E¥@¬öªº³Ì¥D­n¼Æ¾Ç°ÝÃD¤§¤@. ¥¦¤]¬O±a°Ê¤F¼Æ¾Ç¤ÀªRªºµo®iªº¤õ¨®ÀY¤§¤@. §Ú­Ì¥u»¡¤@ºØ²³æªºª¬ªp.

\begin{displaymath}\vbox{\hsize 4.5truein\kai
¦pªG $f(x)$ ¦b $[-\pi, \pi]$ ¤¤³sÄ...
...h¨ä³Å¥ß¸­¯Å¼Æ¦¬ÀÄ, ¦Ó¥B·¥­­¨ç¼Æ¦b $(-\pi, \pi)$ ¤¤µ¥©ó­ì¨ç¼Æ.
}\end{displaymath}

²ßÃD 26


26.1 ½Ò¥» 10.6 ²ßÃD 1-4.

26.2 ½Ò¥» 10.6 ²ßÃD 6, 7.

26.3 ½Ò¥» 10.6 ²ßÃD 19.

26.4 ½Ò¥» 10.6 ²ßÃD 25, 27.

;''


 
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1998-06-27