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标题: [求助]谁能帮我翻译一下这篇文章?~ [打印本页]

作者: angel_lys 时间: 2007-11-23 22:16
标题: [求助]谁能帮我翻译一下这篇文章?~

文章是讲一致收敛和微分的~我英语不好看不明白~请大家帮帮忙啊~~~~

Theorem 9.13. Assume that each term of is a real-valued function having a finite derivative at each point of an open interval 4 L `) y7 h2 H) l . Assume that for at least one point9 v# j7 \$ u8 M in ; J1 I5 C9 O w; [$ B( s7 e- ~5 p the sequence converges. Assume further that there exists a function g such that% h# j y5 t" J9 I2 @ @8 n uniformly on8 j! b) w) [5 G9 d& m . Then:

$ b+ y W" P/ b; S8 y

a) There exists a function f such that C4 _7 {! n" D: K2 l3 B uniformly on' e# A6 W4 Y% B) A C% h ]$ U1 s& K5 K9 b .

b) For each x in / i/ |/ n% { P! a# q5 X+ L the derivative$ r6 W1 X6 V: M' D% \' ^9 ^ exists and equal # x2 a) C) |; ]: B+ e .

Proof. Assume that ! g! e: z4 {4 g' C and define a new sequence 8 J0 m4 H, j7 S7 `7 o4 A8 t as follows:

8 V' w) [6 I3 @

' V$ v6 g8 q5 a+ C* u6 k0 n W (8)

5 t9 D! m& j- }7 G

The sequence4 W& k8 u5 n- T, B so formed depends on the choice of c. Convergence of follows from the hypothesis, since ( g9 g4 O; F: \2 z6 E4 B . We will prove next that( a8 H( }9 f( g( ?& s converges uniformly on$ p2 n: X) d. ~- s& C) B . If , we have

% c5 V$ R/ d9 Y& R

, 3 W; g; j9 }/ z3 X& y/ M (9)

& H, d, X0 {8 }2 p; v

where % }7 U0 \$ O+ `4 _! s . Now 3 c! O# B5 P. f) Z0 G5 \) F exists for each x in ( o$ ~9 Z' A6 L% o a4 d+ c" { and has the value( f5 j& v% j9 x C . Applying the Mean-Value Theorem in (9), we get

, ; O8 }' D' `( I8 P3 @- Y7 D$ I8 J; E 4 g+ I! \# w0 H! K (10)

where7 \2 o- [8 r* e9 E- x% L& x lies between x and c. Since / O1 q; ~& l- i! w p& h converges uniformly on0 t6 j. Q% D# u. J1 z (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that& K' m, n0 J# S* l% v converges uniformly on7 J0 {( s( J* J8 C .

Now we can show that3 O' N+ |- A+ |3 P) k. r. N converges uniformly on ' }4 F3 H+ V8 l . Let us form the particular sequence5 e- t8 O# j6 m7 Q2 J% d corresponding to the special point 9 t2 F3 B4 y4 y5 ?. d for which 3 N& A/ W9 r$ T- W is assumed to converge. Form (8) we can write

an equation which holds for every x in " V0 j; d$ O$ I% `& ]7 x0 K1 m . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on% N# N* M( c% y! E8 R2 x . This proves (a).

To prove (b), return to the sequence3 k8 a0 v @+ D8 e+ u3 x defined by (8) for an arbitrary point c in& B, c ]1 H) t$ a and let8 b% h$ [& V( g . The hypothesis that # j$ R; L) d& }; a exists means that . In other words, each . Q7 C% o6 J4 q% L- l" Z8 P1 J is continuous at c. Since ( W( U0 K: @5 Y( O/ M! d uniformly on - j9 S& S1 b: Q& {% x/ m , the limit function G is also continuous at c. This means that

4 h# O. E- O! q; T9 Y( }( N8 u (11)

the existence of the limit being part of the conclusion. But for + ?( x+ L: G' c* U0 E# g , we have

Hence, (11) states that the derivative8 ?* V/ I ^9 }: o! V+ x$ Z exists and equals i& }( g" k b, v . But

hence k, I$ m: N! l0 o4 @5 k4 h4 V . Since c is an arbitrary point of % e& {# X" z2 q/ A# v/ p4 E , this proves (b).

When we reformulate Theorem 9.13 in terms of series, we obtain

) h$ u, |$ d! z0 Q# m. Q

Theorem 9.14. Assume that each \0 F6 ^( e( f; Y* |1 T2 A% H, J is a real-valued function defined on0 D0 O2 g; I: v( F' W such that the derivative 2 r) s; V7 D2 L( ^1 C! b/ c6 s: } exists for each x in. C; J! l% r6 s. l3 U . Assume that, for at least one point - w* f9 ^. S/ f' C$ n in . T( z9 x% q8 u: @# V( O/ B , the series% y: ?# O$ f6 I# p$ z1 F converges. Assume further that there exists a function g such that (uniformly on " ]' M8 v( m) T4 p$ a: M ). Then:

a)) p. q7 Q! v" S" |" p, S There exists a function f such that 2 @" \7 j2 k3 n8 h6 M- k" q (uniformly on & y) K3 o" m1 b ).

b) 7 h9 M+ {5 L' |) u If , the derivative; ?5 Q2 r$ B. x9 U0 e exists and equals j% @& F" s. K" |/ ~. l .

作者: lzh0601 时间: 2008-4-19 13:22
我怎么只看到脚本文件呢？

作者: tinkertinker 时间: 2008-7-28 14:22
“Theorem 9.13. Assume that each term of is a。。。。。” 这里面有没有漏了字？

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

作者: rao 时间: 2009-1-13 21:47
看不清楚。。。。。

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