[求助]谁能帮我翻译一下这篇文章?~

angel_lys

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 , `6 p5 L, Y6 j4 \8 A# } . Assume that for at least one point in the sequence converges. Assume further that there exists a function g such that P) B: S% u7 R; |: o- j* i- ~ uniformly on % t( ^7 b+ Q6 K9 q3 Z' [0 B' G# G. j8 Q . Then:

9 e6 z* c T0 g5 y' X

a) There exists a function f such that }/ K" b% z; {% ^( |$ d# S! T; ^ uniformly on .

b) For each x in ! f3 }* m9 C) W, N- H! @ the derivative 1 O+ ]8 Z# m0 Q1 I0 d5 V1 a exists and equal .

Proof. Assume that 1 r1 @9 O" j8 W; t4 Z$ t7 Y and define a new sequence as follows:

x& b8 a& x0 M7 o& P) z2 x( |

(8)

# q9 q) @: ?3 Q! \+ @

The sequence 7 `) J5 {; w3 w6 y2 V1 @ so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that ; x& `4 B/ [* f. J5 X0 \1 E converges uniformly on 5 J; l1 S( ?+ z8 l/ ^. ` . If , we have

, (9)

4 [% \. B8 i! s; h$ ]- e

where 2 H. W* T5 n. t . Now exists for each x in 0 x0 \# |$ y& R0 Q3 d; ~$ f% b3 K b and has the value . Applying the Mean-Value Theorem in (9), we get

, (10)

where ' h2 b6 k6 B( q1 j7 L/ w lies between x and c. Since converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on .

Now we can show that ) n6 E* b3 D3 C; u2 Y$ U converges uniformly on ; M3 H& E9 e9 W2 ] . Let us form the particular sequence 4 S% E! y1 S4 k. X) \" Z$ j3 N corresponding to the special point x3 ?' h" A: D" F* O6 [) I for which 5 P6 e8 B+ H5 F/ O is assumed to converge. Form (8) we can write

an equation which holds for every x in 4 x; D$ N6 E5 w6 b& e( F! i . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on . Q5 T }9 S# _1 \ . This proves (a).

To prove (b), return to the sequence & ^' V5 p; j" D7 y0 ~* `$ b defined by (8) for an arbitrary point c in 2 d" |- m( g$ Z6 R7 y1 y$ F' M and let / g; E. v& ~% i" m$ x4 m' }5 H . The hypothesis that . s7 Q6 g1 Y' w. e( x exists means that . In other words, each is continuous at c. Since uniformly on % i$ }! A- @) D , the limit function G is also continuous at c. This means that

# M# z1 g* i# Q' w( n" s; @ (11)

the existence of the limit being part of the conclusion. But for , we have

Hence, (11) states that the derivative 5 i$ U/ `! [9 b! B5 q exists and equals 4 P2 G* T! E( v, H . But

hence ' M! w. v- x( Z9 k8 O: N . Since c is an arbitrary point of . v5 g# S- _4 i1 S , this proves (b).

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

( Y4 F! n7 D6 h5 z4 o1 i

Theorem 9.14. Assume that each 7 F+ j" `& M+ ~) v* { is a real-valued function defined on . d! K8 \4 k& I7 b, H9 o2 h such that the derivative exists for each x in . Assume that, for at least one point in : G& _: Q6 q6 I+ v6 h( u , the series / C P0 X7 h. A6 o2 B converges. Assume further that there exists a function g such that (uniformly on ). Then:

a) 3 ^% w% X+ L5 ]* v. \! O+ ] There exists a function f such that (uniformly on + R0 U+ f8 B6 R& q! z. ~ ).

b) If , the derivative exists and equals " a2 Y& j; M4 k! U7 g .

lzh0601

我怎么只看到脚本文件呢？

tinkertinker

 “Theorem 9.13. Assume that each term of is a。。。。。” 这里面有没有漏了字？
* Z* o4 o, Q' ]! T" m- F5 [

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。