[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 # ?% h& \' a: d. o7 b, h7 V . Assume that for at least one point in 6 S% q: `6 y9 e/ X# b4 Y N8 c* j the sequence converges. Assume further that there exists a function g such that ! J; \5 g( D3 @4 I uniformly on / R# d; S8 u7 L) d . Then:

a) There exists a function f such that - ^6 B/ e1 w4 J uniformly on .

b) For each x in the derivative % v$ y7 ?0 T( k0 { exists and equal ! q7 X$ b' g) ^ l% h/ m0 x .

Proof. Assume that 1 O h1 N$ ~( Q2 z6 i# t and define a new sequence 4 q9 i8 @, g* n L2 d as follows:

4 x! U$ x& n1 z1 C# c( D

/ N$ ? Y- q: {; ~ (8)

|0 j0 _& W7 k/ X% p) x

The sequence 3 S( p0 J- `) [ so formed depends on the choice of c. Convergence of follows from the hypothesis, since : N+ H& R# `* ?& B; [ . We will prove next that converges uniformly on . If , we have

7 G: E. c. c ?+ D* t9 I3 U9 f

, (9)

/ b. q3 \, N5 h6 I1 w5 ?; M+ N

where & L* S: F' n% o$ q# t) z5 d . Now # T* {! A7 N6 p& ` exists for each x in + A; X0 J% b3 E and has the value : f( ^' q! j6 C . Applying the Mean-Value Theorem in (9), we get

, & T' k. G$ M; y$ F2 ^ (10)

where lies between x and c. Since 0 N: J$ w# w/ | converges uniformly on 2 H1 n" d6 b& N (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 . Let us form the particular sequence # X0 c' P+ i, p l& j9 _ corresponding to the special point 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 # h7 y# X" U7 @3 I2 e1 z/ f- f . 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 7 j2 E; Y( W" i# t- l# x% V is continuous at c. Since 3 s1 i; `1 ?( ^6 e uniformly on , the limit function G is also continuous at c. This means that

% f8 |, ~, D: q4 C7 E4 v8 t% t (11)

the existence of the limit being part of the conclusion. But for 3 H/ J1 m6 E* ^" f* X( i , we have

Hence, (11) states that the derivative ) O( x, l) h* O$ w ^ exists and equals 5 J/ [; @; q6 F . But

hence . Since c is an arbitrary point of ! N. P* }" J; I/ ]; q , this proves (b).

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

9 N2 R- h( e1 Q+ h

Theorem 9.14. Assume that each 9 O* E5 P6 |# L& T; I' W is a real-valued function defined on * h0 J' |* T1 l6 u) j such that the derivative exists for each x in . Assume that, for at least one point . K2 v, v" R+ R( I in , the series converges. Assume further that there exists a function g such that (uniformly on ' d! y8 E( F# i& U ). Then:

a) $ H# t; a. Q) j" c( ] There exists a function f such that (uniformly on ).

b) 4 r9 v* Z: Y' h2 T( ^9 ]/ B If , the derivative exists and equals ! K* Y$ ]: d E/ t- ]) H: U3 c .

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。。。。。