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[求助]谁能帮我翻译一下这篇文章?~

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电梯直达

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发表于 2007-11-23 22:16 |只看该作者 |倒序浏览

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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; u5 b8 P! c1 Y9 Y . Assume that for at least one point # b6 @( a4 c. J& b, f/ P- m8 ` in . ? V- W# i. @$ W4 W the sequence converges. Assume further that there exists a function g such that " [7 p0 X/ ?0 {4 W5 f uniformly on 0 I- N- `5 O6 b# h' }9 s . Then:

- Y7 m, i" h" x. z

a) There exists a function f such that9 C; W C& o, n8 D uniformly on' l5 j) y& f2 c5 q .

b) For each x in+ _' r3 b) P8 ?" m4 U3 P. \ the derivative , {' F; e6 N$ H f1 {* ` e! A exists and equal 9 I5 J: t7 d( Q: q; |, d) v2 b2 N .

Proof. Assume that" _2 z$ B5 ?. G' d, \ and define a new sequence * Q4 h3 z0 u7 h5 d1 Q8 L$ ` as follows:

5 g" X' D6 i1 r( c

$ K! C5 N* k. O1 } (8)

J+ w h. D6 y P4 v+ {) P8 ^0 t/ B( s

The sequence2 e' y! s" ~( o. r! L2 W6 j so formed depends on the choice of c. Convergence of follows from the hypothesis, since ( b" U: X+ G4 }. c7 h: l+ u . We will prove next that6 O, J. Z* k/ t- P$ Y. c converges uniformly on9 Y. S$ s3 V3 {! K$ m; n: y/ Y5 X . If , we have

4 b4 y0 X7 w9 ]- S6 a) q }

,* ~8 R5 q \" P8 I; Q. v3 V (9)

% N% C- p2 c& P& T

where 5 t! l: S# U# C7 ? . Now: J5 H1 F7 R' t- R! A8 H$ u. d exists for each x in 1 f8 G1 d1 }4 b' ? and has the value3 M3 \) ~7 C, I! \ . Applying the Mean-Value Theorem in (9), we get

,! W8 Z# ~: H6 J 5 i- Q4 }$ J X/ b8 ~ (10)

where1 l+ @9 P6 h- R# b4 D( k8 W lies between x and c. Since 2 M c; d+ n/ t& G% F converges uniformly on F& P/ n1 p, w+ `, O (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that J5 i8 `( r+ `! R( _2 {+ p0 d! d converges uniformly on 4 S' ?% ]" H4 _ .

Now we can show that# u: L* @" Y" {9 ?& g2 Q+ m converges uniformly on( M" c+ E* N! H . Let us form the particular sequence % Y% j4 m; H9 e corresponding to the special point 2 I1 i7 L& C/ d; o' l# @ for which/ W( C* c. s& ?6 d" @5 q/ k is assumed to converge. Form (8) we can write

an equation which holds for every x in 9 n3 Z8 b0 v9 s; h . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on 3 _6 | @8 `) B% ]2 _ . This proves (a).

To prove (b), return to the sequence 6 a- W' T5 c/ m defined by (8) for an arbitrary point c in" F* M1 w1 o: m, I) u. h and let * A; i7 M. J- s% p2 a0 A . The hypothesis that # f, f/ h5 k$ a* o exists means that . In other words, each9 ^% T; O) K2 i is continuous at c. Since ) a9 X3 U4 o) ~4 h( H uniformly on' g$ I5 ]: R/ S( t! t , the limit function G is also continuous at c. This means that

! [/ V; [$ }9 X: M' `7 q (11)

the existence of the limit being part of the conclusion. But for+ p% n4 x' \' N' O% ~1 X , we have

Hence, (11) states that the derivative: n& }+ n; g) J, A. q0 q exists and equals * |- ?+ ~7 t2 b; L9 _- I+ o4 | . But

hence- r2 u% w1 y0 \9 _ . Since c is an arbitrary point of2 K* a6 F1 J& D5 z/ w; k , this proves (b).

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

}1 L* Q. l- I( ? k

Theorem 9.14. Assume that each 1 {+ ?' Z* X8 U7 V4 W# I+ j1 \( G is a real-valued function defined on 3 Y: k3 b7 A1 X; @$ a9 v such that the derivative; v1 u+ i n/ s& s. |# @7 W! d& r$ a exists for each x in ) n* w+ u2 d+ _ }) I) c* f) O . Assume that, for at least one point7 O* h- q8 j( A3 H( R in 2 {1 @ c0 L# L , the series( d* t2 F( v% m converges. Assume further that there exists a function g such that (uniformly on9 l( x9 p8 e( M8 m2 b: J ). Then: