[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 . Assume that for at least one point 8 f( `6 q; P1 T" f( K6 W in ! f$ | S6 v! P) K. y the sequence converges. Assume further that there exists a function g such that uniformly on . Then:

) [' L) f; `+ q0 m" k3 U

a) There exists a function f such that ! j/ ?: S7 F/ e* e0 ? uniformly on .

b) For each x in 2 P" M! L: D/ q3 ?5 e the derivative exists and equal .

Proof. Assume that ! P: w/ f/ Z. R7 m- y and define a new sequence as follows:

(8)

The sequence : ]4 }) T" T$ @ so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that converges uniformly on 8 X6 b: O: X. X& K% f . If , we have

, (9)

where . Now exists for each x in ) ?5 d2 o) s6 i& ^% X and has the value " s a& e, Q! { k . Applying the Mean-Value Theorem in (9), we get

, 4 @# |' J3 p2 n% K, D2 V (10)

where lies between x and c. Since / [! K. q) f' G" S% J- U0 e# O( ` converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on % u6 g$ I" B6 h$ g! n j9 E' [# j .

Now we can show that converges uniformly on . Let us form the particular sequence % ^3 o* Z, N* U+ X. K& ~- j: O corresponding to the special point for which 8 P! f l3 |1 n is assumed to converge. Form (8) we can write

an equation which holds for every x in . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on . This proves (a).

To prove (b), return to the sequence & {5 J& E% J, a( j7 z1 D0 E defined by (8) for an arbitrary point c in * z8 L+ @( \% t) I and let ! \8 z8 Y0 c: ~ @; e5 W/ e* o( s . The hypothesis that exists means that . In other words, each is continuous at c. Since ( U3 u2 P8 D2 Z. A- |9 n uniformly on : c4 y3 K' G3 F$ Z , the limit function G is also continuous at c. This means that

5 ?* f# K6 N- j$ ]; h) r (11)

the existence of the limit being part of the conclusion. But for , t6 L) ~! |$ p- A9 D , we have

Hence, (11) states that the derivative . M7 r) B. ]+ d; J7 ] exists and equals / m; V( I4 m1 v . But

hence . Since c is an arbitrary point of , this proves (b).

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

0 B, ^. O$ c' y# p

Theorem 9.14. Assume that each / L* X9 p& ^' F+ c is a real-valued function defined on 0 I0 d/ g& H" } such that the derivative ! Y' A$ K1 @2 U+ ~3 T0 U* E exists for each x in " S. e% Q3 P3 W' f" f+ \) ~8 ~ . Assume that, for at least one point - ]: f4 m9 M/ ^' ~2 _" r. P6 l in , the series 1 q. x5 ]' N0 O4 H; U converges. Assume further that there exists a function g such that (uniformly on 4 N! |2 w6 W a/ _8 p# M ). Then:

a) There exists a function f such that 9 C& O5 R( B7 A* T (uniformly on * t7 w1 j/ k) _# o u% d& x ).

b) 0 s: o* r7 y6 Q5 \9 N8 u If , the derivative exists and equals .

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？

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。