[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 - Z( u% [! I! {! K . Assume that for at least one point # J1 n4 f: m' t) z in the sequence converges. Assume further that there exists a function g such that 0 o9 q3 z$ u0 @# y' Y* K uniformly on . Then:

a) There exists a function f such that 4 B- t/ m4 D6 q0 m- f/ G uniformly on 2 ~+ G4 h' w( D! X .

b) For each x in the derivative exists and equal .

Proof. Assume that and define a new sequence 3 V% Q( j4 m" e& C1 F! S as follows:

(8)

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that converges uniformly on 8 |$ W% b5 H: J4 B5 U3 }' O4 t: v . If , we have

& Z. t0 o5 I3 h5 n8 S p6 [1 z

, (9)

where ( V: o! n$ m( z . Now $ S# h+ ], M8 u5 ^ exists for each x in % a3 T4 F/ |6 Q+ o1 m# B2 Q and has the value . Applying the Mean-Value Theorem in (9), we get

, (10)

where lies between x and c. Since / t! E& v E4 j S" w1 I! t0 j converges uniformly on 9 |' t9 |0 `, A (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that ' z' D0 }. _* b9 A4 J converges uniformly on % p4 I. `/ X$ p .

Now we can show that - s& s& T3 @3 M: D converges uniformly on 0 w0 b" X( \ Y9 v- f3 J . Let us form the particular sequence 7 B$ ^, N; ^: Y: l# \6 p corresponding to the special point for which . M3 M$ c; y+ D6 d is assumed to converge. Form (8) we can write

an equation which holds for every x in / T8 `' k" f- c- w: \+ n! T4 p# o Z. g . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on : G U3 p; P; ?' c' E . This proves (a).

To prove (b), return to the sequence defined by (8) for an arbitrary point c in and let ! g; A/ [1 {, D: m' w . The hypothesis that exists means that . In other words, each is continuous at c. Since - N! m! z% |7 ] m7 v4 z% n! Q 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 " t3 H5 g6 Q/ w8 B1 Z ^0 ?4 V exists and equals . But

hence 0 R. S4 p5 p4 Z- y4 Z . Since c is an arbitrary point of , this proves (b).

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

Theorem 9.14. Assume that each is a real-valued function defined on such that the derivative exists for each x in ! O2 X* O" [; M" Z2 T4 ]- f( p4 G . Assume that, for at least one point in , the series 7 b O: E) S% P; a6 _2 D3 z converges. Assume further that there exists a function g such that (uniformly on ) `9 y, s. n5 I9 ? ). Then:

a) + ^1 M& H; V: v$ r* Z5 _4 Y6 V5 E4 P There exists a function f such that & h( P- d, h7 F% i (uniformly on 8 |! [6 h2 l; h+ U ).

b) . J. [: ]' {' ~; N1 ~+ c If , the derivative ) J& f3 n- O3 k" K, u9 ~+ l exists and equals .

rao

看不清楚。。。。。

tinkertinker

 “Theorem 9.13. Assume that each term of is a。。。。。” 这里面有没有漏了字？
3 ~' E6 c: h+ z4 J, l

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

lzh0601

我怎么只看到脚本文件呢？