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Next: 2.4 $BIUO?(B 1: $B@~7A:9J,J}Dx<0(B Up: 2.3 $BGHF0J}Dx<0$N=i4|CM6-3&CMLdBj$N:9J,2r(B Previous: 2.3.6.0.4 $B>ZL@(B

2.3.7 $B:9J,2r$N87L)2r$X$N<}B+(B

$B$5$F!"(B

$\displaystyle \sum_{n=1}^\infty \vert f_n\vert\DefEq M_f <\infty,\quad
\sum_{n=1}^\infty \vert g_n\vert\DefEq M_g <\infty.
$

$B$r2>Dj$9$k!#(B $B$9$k$H(B $ \forall \eps>0$ $B$KBP$7$F!"==J,Bg$-$J(B $ N\in\N$ $B$r

$\displaystyle \sum_{n=N+1}^\infty\vert f_n\vert<\eps, \quad
\sum_{n=N+1}^\infty\vert g_n\vert<\eps.
$


  $\displaystyle \left\vert u(x,t)-v(x,t)\right\vert$ $\displaystyle =$ $\displaystyle \left\vert
\sum_{n=1}^\infty \sin n\pi x
\left(f_n \cos n\pi t+g_...
...t(f_n \cos\beta_n t+g_n \frac{k\sin\beta_n t}{\sin\beta_n k}\right)
\right\vert$
    $\displaystyle \le$ $\displaystyle \sum_{n=1}^\infty
\left(\vert f_n\vert\left\vert\cos n\pi t-\cos\...
...ac{\sin n\pi t}{n\pi}-\frac{k\sin\beta_n t}{\sin \beta_n k}
\right\vert
\right)$
    $\displaystyle \le$ $\displaystyle \sum_{n=1}^N
(\vert f_n\vert\left\vert\cos n\pi t-\cos\beta_n t\r...
...ert\frac{\sin n\pi t}{n\pi}-\frac{k\sin\beta_n t}{\sin \beta_n k}
\right\vert
)$
      $\displaystyle +\sum_{n>N} \vert f_n\vert(\vert\cos n\pi t\vert+\vert\cos\beta_n...
...right\vert
+\left\vert\frac{k\sin \beta_n t}{\sin \beta_n k}\right\vert
\right)$
    $\displaystyle \le$ $\displaystyle \sup_{1\le n\le N\atop \vert t\vert\le T}\vert\cos n\pi t-\cos\be...
...pi}-\frac{k\sin\beta_n t}{\sin\beta_n k}
\right\vert\sum_{n=1}^N \vert g_n\vert$
      $\displaystyle +\sum_{n>N} \vert f_n\vert(\vert\cos n\pi t\vert+\vert\cos\beta_n...
...right\vert
+\left\vert\frac{k\sin \beta_n t}{\sin \beta_n k}\right\vert
\right)$
    $\displaystyle \le$ $\displaystyle M_f \sup_{1\le n\le N\atop \vert t\vert\le T}\vert\cos n\pi t-\co...
...t\vert\frac{\sin n\pi t}{n\pi}-\frac{k\sin\beta_n t}{\sin\beta_n k}
\right\vert$
      $\displaystyle +\left(2+\frac{1}{\pi}+T\right)\eps$

$ h$ $B$r==J,>.$5$/$9$l$PBh(B1$B9`(B, $BBh(B2$B9`$NOB$O(B $ \eps$ $B$G2!$5$($i$l$k!#(B

$\displaystyle \vert u(x,t)-v(x,t)\vert\le \eps+\left(2+\frac{1}{\pi}+T\right)\eps
\le \left(3+\frac{1}{\pi}+T\right)\eps. \qed
$


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Next: 2.4 $BIUO?(B 1: $B@~7A:9J,J}Dx<0(B Up: 2.3 $BGHF0J}Dx<0$N=i4|CM6-3&CMLdBj$N:9J,2r(B Previous: 2.3.6.0.4 $B>ZL@(B
Masashi Katsurada
$BJ?@.(B14$BG/(B11$B7n(B29$BF|(B