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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% K4 k0 S* Z% z# i9 J/ r . Assume that for at least one point2 n! F" r7 O- U; O3 |; S2 R in3 u$ T K$ E5 A" g the sequence converges. Assume further that there exists a function g such that4 h- G' K/ w$ M+ T6 P5 \5 Z0 N3 L4 k9 w" E uniformly on. v8 X6 r* ?- C5 t . Then:

8 r9 N/ ~# @" R) S

a) There exists a function f such that # d4 [2 G8 \: h7 a. u uniformly on, [" J- s6 h+ i/ x .

b) For each x in 9 E0 H2 s' L- _5 [! A& n the derivative" p! _# t* z) @ G) T8 F: z exists and equal + W) t0 F: s; _$ \6 E. y G .

Proof. Assume that e& Y8 s2 i1 } and define a new sequence " W" n6 R4 f9 @* D) c! m* e; X as follows:

8 V' \3 {9 [/ E- F% m

& D* `# i' A* K0 F7 @ (8)

. j7 B" G4 H0 Z9 w& q) w

The sequence+ P5 V9 s L' h so formed depends on the choice of c. Convergence of follows from the hypothesis, since ) n0 d/ l2 u4 V . We will prove next that 9 _% Z0 j6 H" K3 F" Z converges uniformly on* G7 F* g) ?( Q) r$ I( y8 G; t . If , we have

& y# L6 [9 k. v+ ~

, & \* z5 _; b1 V( | (9)

5 s- |5 T, W1 M4 P! w% Y. V

where& x, Z& w! F G( o . Now $ d+ k/ Z3 T) q9 |5 q# T) a1 [: V7 x exists for each x in7 b+ \8 [0 E6 @) w) l and has the value 9 F& \! p4 W/ r1 Z& y$ B# T( A: {% V . Applying the Mean-Value Theorem in (9), we get

, ' n+ n- d: C/ \6 U& L; d . ?8 c9 M& g4 z$ d, h$ g9 u (10)

where : c. Z3 _" d4 Y lies between x and c. Since3 `) Y; k/ b3 C x$ ` converges uniformly on ! U1 x% j* @4 G (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that$ R! y8 q, b+ k6 o) p; d converges uniformly on' o5 e K/ H3 j/ H2 m .

Now we can show that4 T" J1 B. J, x3 k6 q2 N converges uniformly on; N( c; ]- J) Q% A . Let us form the particular sequence# U$ A% ?- @ [% u corresponding to the special point- K/ f' m. D* ^ for which 4 y G3 Q! o" _2 v7 j; T0 x is assumed to converge. Form (8) we can write

an equation which holds for every x in 4 j$ N; s+ t3 \8 ^ . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on% ~- {3 g: {9 M6 @& [7 T% ?. I . This proves (a).

To prove (b), return to the sequence8 h; u$ U7 S+ `% L defined by (8) for an arbitrary point c in: V7 e( m/ U" d0 J2 j and let 9 P f1 H0 Z: o* m) X- i . The hypothesis that 3 j% E4 X% H; k- i4 X; }# l exists means that . In other words, each$ y6 `) J' w3 |2 z' Y is continuous at c. Since $ E' Y" {& d8 R% _: e, N% o$ A% d D uniformly on 5 P6 E7 ^- S/ x2 z , the limit function G is also continuous at c. This means that

( B9 O+ m6 K' l; I (11)

the existence of the limit being part of the conclusion. But for. O# A/ w" A4 x) A9 k) F9 X , we have

Hence, (11) states that the derivative1 B8 V y" P% `# X: B& M exists and equals8 b, `, W" u0 M+ \" G. n, ^9 w . But

hence8 K @, n8 A; U5 ]2 K* ^ . Since c is an arbitrary point of( C* j: t6 D4 |3 O , this proves (b).

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

/ h( y R( k7 G6 N, b, g' k

Theorem 9.14. Assume that each ) w5 |8 c# m4 l. A+ ?, V is a real-valued function defined on/ F: h* k& W4 {; u such that the derivative " h2 e' t( L8 ~ exists for each x in1 b- m+ ~8 y3 A5 I6 d. U . Assume that, for at least one point* p2 M8 c* j0 ?5 l. y in. k' r' Y* b5 O! q8 u1 ^2 s. b , the series2 T0 ~8 o4 x) _5 F* M converges. Assume further that there exists a function g such that (uniformly on( v! M% ]* i" f+ l) I4 L ). Then:

a)7 p* v4 o- R- a, U K5 Z There exists a function f such that " H) `' H" K5 \/ C (uniformly on( f6 h9 h; A1 A/ X% m ).

b)3 i( s4 B8 w5 `3 { [' I4 q: d If , the derivative! D; |6 Y M; w& N( k exists and equals5 y7 ?& ^, ^* Z& j0 ^ .


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?
7 R/ _2 W7 W9 h/ R
[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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