[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 - R1 f# v/ v* h& k k in the sequence converges. Assume further that there exists a function g such that # A2 Y" c" n5 j# r( M uniformly on . Then:

" W( L* e+ I( D* A+ l* ?9 Z

a) There exists a function f such that uniformly on ' G2 r: S6 R8 ^. e0 P! d& v' Q2 s .

b) For each x in the derivative ! s3 _1 {$ p* x exists and equal , W$ H# Z! K% m. a .

Proof. Assume that # [! L, i7 N p* O7 p and define a new sequence & {' @: j# T8 D+ I8 m# b2 E# M as follows:

. a+ \ V- X; c- V# a

(8)

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that " Q2 i9 V7 U6 L1 J; c converges uniformly on / C, f) d' D$ x . If , we have

3 `0 k1 b) `1 y/ Y! m" u" Y

, / G- f) u0 n+ t (9)

where . W2 _( ^) d+ i4 @8 q3 E7 v# P, d . Now & p$ e; W7 {, T- U. ~ exists for each x in ; f; U+ C5 b) f$ E and has the value . Applying the Mean-Value Theorem in (9), we get

, * h' R* l2 {0 \ v }7 v$ w) A (10)

where / C# z% y! K' @ lies between x and c. Since converges uniformly on H' M; U- N& W$ x5 U (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that / H8 X" f% A; [( D, r/ V: V converges uniformly on .

Now we can show that * x) G3 d! y4 D6 h1 B# S. f converges uniformly on + h8 {6 ]2 o, l! F0 J . Let us form the particular sequence ( }/ I# D* C1 r corresponding to the special point ; [+ S1 O% j5 E3 v) I for which ( q2 @( E$ R, z is assumed to converge. Form (8) we can write

an equation which holds for every x in 4 m# ^' E2 [$ Q" t [& u . 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 # h. v% u5 M% ]5 P3 N# k and let . The hypothesis that exists means that . In other words, each is continuous at c. Since uniformly on * X/ i4 B6 e- Q# C; y9 U/ S , the limit function G is also continuous at c. This means that

+ i( Z8 w$ B" o. Z! U+ ? (11)

the existence of the limit being part of the conclusion. But for 7 ^7 K( F! _4 w9 b( n4 ~( Z" N , we have

Hence, (11) states that the derivative $ b* p6 C' `* r- V exists and equals # m1 A6 X. O! T . But

hence ! }2 A& |5 F5 a% |+ k& K1 h$ O . Since c is an arbitrary point of , this proves (b).

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

( U& j# d) E4 G

Theorem 9.14. Assume that each is a real-valued function defined on 0 x9 H: x' Z0 G, c+ }0 q+ U# n7 X such that the derivative exists for each x in 8 u' {* p& _$ B; P9 {9 o' X7 x . Assume that, for at least one point 8 g: {$ g3 a% j in , the series 4 e# W3 C! }& q" |3 T converges. Assume further that there exists a function g such that (uniformly on ; M. C% ~# w9 q+ F# g6 R4 g ). Then:

a) 5 b( m) a- E5 B/ B There exists a function f such that }* \+ q2 x$ b8 A (uniformly on ).

b) If , the derivative exists and equals # U R9 c' [- v. [- a+ ~ .

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？
' {8 N* O( {. M: Z

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。