[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 * q/ g( I" _4 w0 ?# h* @+ P . Assume that for at least one point in the sequence converges. Assume further that there exists a function g such that $ ? d; h) i5 V uniformly on * B. n" k0 q& v . Then:

a) There exists a function f such that uniformly on .

b) For each x in the derivative exists and equal 0 \$ |8 v. o4 c# W3 E) J .

Proof. Assume that ; e0 n7 V+ M9 Y0 g0 e6 U and define a new sequence ' X2 u5 `! u7 r9 y8 h6 w as follows:

(8)

& G: J( u* A$ O4 u1 {

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since + h# ^2 C. a9 |' U d& t2 w . We will prove next that 0 h1 A2 F, ?3 x) N converges uniformly on . If , we have

: Y6 y. W* a+ B8 M9 W) n

, (9)

. V; j) S; U4 B8 r2 p$ m

where $ [8 D0 b9 B4 x2 c0 T6 z5 Z . Now ]/ t3 p. E* d: q3 b+ x1 E exists for each x in and has the value . Applying the Mean-Value Theorem in (9), we get

, (10)

where 0 @, k$ P i) V& H8 T, N lies between x and c. Since % H/ m) P8 |% _! p& ? converges uniformly on 8 p& Q' c9 ^+ {5 B# V. e# F (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on .

Now we can show that converges uniformly on % ` S- X3 B! m' D, ]9 e . Let us form the particular sequence corresponding to the special point for which 4 u- f9 c, U- L, Q$ b4 T" @* b' U/ Q 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 2 E) }1 b) t0 w$ m . This proves (a).

To prove (b), return to the sequence defined by (8) for an arbitrary point c in 7 j, P$ R, `/ N and let . The hypothesis that exists means that . In other words, each # L& I$ H1 C# e, ]! {. }, N, N7 k is continuous at c. Since 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 8 D: J& Q0 l; ?& j" J" d# Q" p , we have

Hence, (11) states that the derivative exists and equals 2 D1 ~4 H$ M6 L | . But

hence 8 I0 Z% o2 m C% i5 A, U# |5 w% ^ . Since c is an arbitrary point of o4 [. ]% `7 l8 r( \ , this proves (b).

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

Theorem 9.14. Assume that each / _7 N* z, a! H- ^% J t: { is a real-valued function defined on & u7 ^& J# r0 \3 @9 _8 R8 Z such that the derivative exists for each x in . Assume that, for at least one point in 5 V( K) u, Z) Y; ~ , the series converges. Assume further that there exists a function g such that (uniformly on 1 }! i' j! m$ p9 v! Y ). Then:

a) There exists a function f such that , U/ g- U) r- s; r9 p; h (uniformly on ! }- d! q! F- S- p/ _8 J' F: V ).

b) + ^; d9 g7 O6 ] If , the derivative exists and equals ( m) X3 w: b8 ]% s/ ` .

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？
( F% w4 X; b( D

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。