数学建模社区-数学中国

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 ; v* Z' Y! `0 c2 a . Assume that for at least one point; K3 }1 ~: F, [ in) I& e- {( t( u' N. A+ }% z- Z, l7 z* b6 S: ? the sequence converges. Assume further that there exists a function g such that / P* k) g8 n: B( P D% s uniformly on; Q: L h! d: H7 N8 d . Then:

$ L: J) i: G+ B% X

a) There exists a function f such that 7 n9 Y' t& M+ v; [ E0 @ uniformly on ) T3 b# n0 ^: T r( Z& u; c .

b) For each x in5 w: x! [! _' q% H# [$ t: ~ the derivative + d6 ]5 A0 ~$ E, W exists and equal ; M' i2 |- x: y8 m) D. \ .

Proof. Assume that ) l/ `% O; a& f5 A) k2 } and define a new sequence 1 ~ z! b7 [+ J as follows:

# g( u* ~& h* \" p5 ]# y C

8 k, @) v2 s* M8 p& N/ X (8)

% @+ N1 r; g9 f/ `# q! d1 T

The sequence 3 G$ @' P! X, x. K0 [$ P* f' s% k5 P so formed depends on the choice of c. Convergence of follows from the hypothesis, since 0 e) w. f. d4 ~6 P& u( x . We will prove next that) J2 C2 C" F$ `9 _" R$ b( F, u converges uniformly on 7 [' ?' p, [! `% ^* u { . If , we have

+ }, d' v( ]9 n9 V

, 3 k( p+ v7 y- {0 k x0 l' ?! I" q (9)

3 l* B& v( }; [2 R9 \% |' i+ ?

where % H) y9 V5 C! ~3 N8 \ . Now& U$ t: z6 H7 h/ A) o exists for each x in 9 _+ Z+ E. n; z& k; }5 w and has the value# [' b* P5 K3 l . Applying the Mean-Value Theorem in (9), we get

, 7 R% C7 L: o' R" W, G3 } 6 }+ o% Y4 l+ A$ h+ y+ i M; {! b% ] (10)

where0 m0 P0 z6 F6 S1 ~* i1 y# w0 V lies between x and c. Since4 L* ?; t8 q: ~. X/ k converges uniformly on 9 Y1 G1 h3 u, D$ Z. V- w, Y0 ? (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that4 t/ M- \0 J* F, t1 \ converges uniformly on # p+ c7 R; d1 m+ p9 |5 y+ \5 U. H7 m" O .

Now we can show that 5 S' V+ ?/ u4 X converges uniformly on2 F* j/ Q0 ~' ]" D, Z3 ? . Let us form the particular sequence * K6 h H9 K) i4 w5 l corresponding to the special point7 N q+ W' F$ S- g ?. J; V3 ] for which( r# {* |3 q; A2 P4 {# m0 ~ is assumed to converge. Form (8) we can write

an equation which holds for every x in + V9 \; k4 M& p/ ~ h, N- E; I . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on* i" E: x6 J) o; s3 d . This proves (a).

To prove (b), return to the sequence 0 Q3 t% Y7 V e1 @, i P defined by (8) for an arbitrary point c in z0 m% q5 j1 y: M" H and let% t6 `$ K! y3 w. E . The hypothesis that $ E. x# R/ H# R" H' r exists means that . In other words, each 3 K2 q4 E$ z3 p' R1 t is continuous at c. Since, y7 l! q; m8 H; q uniformly on 6 T/ R- P* d7 ` , the limit function G is also continuous at c. This means that

l% D, b7 H. g# Y (11)

the existence of the limit being part of the conclusion. But for; r4 C: q- l/ E6 k9 D , we have

Hence, (11) states that the derivative & l8 v& E& n% M0 Z exists and equals / Z3 y( g5 Z/ _. i- { . But

hence % N" M. m1 Y9 W: Q( ]* ~. l . Since c is an arbitrary point of 2 L0 M/ f2 f2 g; L' U2 J) h& A , this proves (b).

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

( n1 k- I1 _# P/ k6 l1 W$ v" `

Theorem 9.14. Assume that each " w: X. J2 D+ W1 `% h is a real-valued function defined on 4 s5 b. z: i8 j1 I3 d; u% i such that the derivative ( p. A7 B9 F7 u) E4 k exists for each x in5 k" w6 a# \5 D( i . Assume that, for at least one point% j4 ?1 o+ l) d& Z+ b/ y+ w in, H6 J$ g$ y) I* c% K" \ X& h , the series " t9 u1 g8 N" u$ N0 g, t converges. Assume further that there exists a function g such that (uniformly on 1 j$ A& W: H% \9 I ). Then:

a). Y8 @: x3 l3 {9 L There exists a function f such that) S) N8 z$ e5 \/ q1 n+ E (uniformly on+ B5 \+ ]! N$ E7 H; K ).

b) ! I# w5 ^; G4 D- d If , the derivative ) p4 l6 E! E% p1 D$ Y# L0 p% V7 Z8 c exists and equals/ F7 Y. ~8 E% V0 X9 } .

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。。。。。

欢迎光临数学建模社区-数学中国 (http://www.madio.net/)