[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 ; m; f! M3 @" _ a . Assume that for at least one point 9 z! O4 \( b% P% f* b5 }% ?5 v. t in the sequence converges. Assume further that there exists a function g such that ( ]( i, L/ k3 u% j uniformly on . c6 k L! G0 V& J' Y8 ` . Then:

a) There exists a function f such that uniformly on ; r( V2 n+ A- {, a& p .

b) For each x in $ c3 P, Z* Z1 V0 O' h# V/ Y! o+ @8 s; l2 t the derivative exists and equal .

Proof. Assume that and define a new sequence as follows:

9 p/ Q8 W4 q8 K) v (8)

6 d9 i1 i. z7 r2 t/ w& O

The sequence 2 k8 L5 M7 \" `5 |( D4 ` so formed depends on the choice of c. Convergence of follows from the hypothesis, since % h2 l0 ]) a9 Q) I . We will prove next that |& I' z3 H+ v; u- _' x converges uniformly on 9 H8 I, s( V( ]* e . If , we have

, (9)

% T4 l5 n N& _/ A- t5 K5 L

where . Now ) U O8 B( ?" }% R: P5 e exists for each x in and has the value . Applying the Mean-Value Theorem in (9), we get

, 7 K: e8 p# g0 x& D/ b& k (10)

where ) R& F @" m9 c+ | lies between x and c. Since & m) F" U5 l2 `0 Q7 \ converges uniformly on * W) n& f- @) v (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that 0 i- P2 t4 v# r/ d; b converges uniformly on .

Now we can show that : L; W; k0 I2 [- C5 |5 ` converges uniformly on a) l" h$ b0 T: i/ o . Let us form the particular sequence corresponding to the special point 9 W8 O- ^1 E. P( O4 b( y, K! [" p for which 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 defined by (8) for an arbitrary point c in and let . The hypothesis that exists means that . In other words, each is continuous at c. Since , O! ^: l- o2 d/ J# j5 G uniformly on , the limit function G is also continuous at c. This means that

(11)

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

Hence, (11) states that the derivative 2 A3 o* W2 C; m exists and equals . But

hence 4 j& e9 a- d: \1 T6 ?4 ^ . Since c is an arbitrary point of 7 K9 K8 r1 `' s6 s, r) V1 h+ q, r , this proves (b).

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

Theorem 9.14. Assume that each ; F" f: d4 {7 V' y5 c is a real-valued function defined on such that the derivative exists for each x in . Assume that, for at least one point in * E# J* e' S* x ^- M3 O , the series converges. Assume further that there exists a function g such that (uniformly on 3 H) Z% m. n0 V# I ). Then:

a) , M. N7 d1 w! Y' i. K; r+ C; M There exists a function f such that . r. W; f7 A3 J$ m7 [ (uniformly on : C3 s* o/ |5 s+ c& O- n6 M ).

b) 3 s0 r3 V/ @9 v; ^4 s8 F If , the derivative exists and equals + V0 n4 Z1 {6 s/ K .

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。。。。。