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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 interval9 S1 Q" q+ M! b3 T" n! ^! t . Assume that for at least one point + V& u9 J5 _! P in9 {+ @1 ` x4 P/ E' A the sequence converges. Assume further that there exists a function g such that ! n5 A' j5 X) S uniformly on% Y; s4 e9 V8 R1 b5 w% H9 _4 g . Then:

! V. A: g( v+ r$ Q5 G0 i

a) There exists a function f such that 7 T" m$ ~( {/ v% m G v; d8 j4 n uniformly on 1 }6 {: X- E$ _% c8 v/ {* L/ L .

b) For each x in the derivative 4 D. `1 \. g. i* ]( p# f exists and equal d' y* b1 l C. ]% F* s .

Proof. Assume that! }7 W; @: g- U9 | and define a new sequence ! T0 ^! r1 w% G- s/ X* O as follows:

9 Q4 b0 l8 F+ W* J* Y

1 _0 w* Y' B) H; I, b% j: ? (8)

, q2 _5 H5 N+ |8 Z! [

The sequence # }9 o8 S$ W' E; ^6 N2 w( U2 _. E so formed depends on the choice of c. Convergence of follows from the hypothesis, since $ b3 f! J Z7 q8 j" c6 k( p% _. n . We will prove next that 7 N( I5 `7 }' r9 r) y8 Q+ w9 e converges uniformly on $ z k/ X: ]8 u; I1 i . If , we have

8 b( V2 k/ b2 e8 l B

, ! ]- S, h ~, v! U/ ^, x' t (9)

1 `, |. z+ D7 `' G8 G8 Z; }0 [

where . S' j9 Q2 f* Z . Now2 M1 q& @% @' _0 u exists for each x in/ l6 P( T- X/ t' X( p; {% \7 `7 r and has the value: }7 O- A# M+ [; |4 E1 b- Y8 h . Applying the Mean-Value Theorem in (9), we get

,, c/ k F2 M$ O8 ? " O; I0 i1 A& R4 ] r9 ] (10)

where& f! [( j I7 R2 ^ lies between x and c. Since 9 n6 {2 V0 ^5 |5 k$ W converges uniformly on 6 `- O w: D2 U (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that , ~7 |" v$ i, L& j4 e converges uniformly on 9 o; c- {9 _ _; k- D, E .

Now we can show that1 w8 P4 n9 t* Z- i8 V* j converges uniformly on: ^/ {8 e& Y: T+ o; I . Let us form the particular sequence $ W4 B! T9 y, k+ O6 ^$ L corresponding to the special point1 F: f2 X1 X' S+ g for which" m5 ?. X6 V. s. O% K6 \ is assumed to converge. Form (8) we can write

an equation which holds for every x in v( W7 |4 x8 W0 c. c! W . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on+ |2 i9 o$ @4 b4 ]2 q . This proves (a).

To prove (b), return to the sequence# o7 a: S; |* d- w defined by (8) for an arbitrary point c in5 o4 N1 H* S: }" F and let 0 t( W2 c4 \2 Y* z* {' |2 f3 U: f$ e: M . The hypothesis that8 C$ [. Y9 D3 }5 V& w$ H exists means that . In other words, each& W) L% B- b8 k is continuous at c. Since; C+ j9 h# g' E2 v! Y" V; @; [ uniformly on9 j, ]$ }; y- F+ E , the limit function G is also continuous at c. This means that

) w. b% s" ^# Q R (11)

the existence of the limit being part of the conclusion. But for, E% w: r- w" F% y , we have

Hence, (11) states that the derivative- v9 A% T( m2 v4 Q exists and equals7 C, [+ |- W4 b- S2 ]5 J; z: K . But

hence* M8 K% p4 D# ^$ g5 B) o . Since c is an arbitrary point of0 }2 ^% Q, O" ?( ^5 L2 _; y , this proves (b).

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

& k7 Q0 X! R/ N! F. f

Theorem 9.14. Assume that each' u3 X$ E; @5 v4 H3 R2 Z6 r is a real-valued function defined on% G2 ^5 p' N( K4 W6 ^2 w such that the derivative; Y% @2 _- ]) ^9 K( j. g( b exists for each x in* j: {! s5 X5 _2 h- T C* J . Assume that, for at least one point" _; F$ M: r) b7 f1 S: x in * _7 ]% ?% a7 Q$ e , the series / A) t& o1 Y z* l1 p converges. Assume further that there exists a function g such that (uniformly on6 \ F: \' e; g Z ). Then: