[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 in " v! d7 z; G0 R& D9 m the sequence converges. Assume further that there exists a function g such that 2 j; u4 J. U7 t7 e. j uniformly on 5 \$ M) ~) c' G1 X4 z& n . Then:

% e) N2 T* V o# y0 W, i

a) There exists a function f such that uniformly on ! }7 `; v+ Y2 K& p6 n: J .

b) For each x in the derivative 9 G: y; C) N/ Q, F1 ~' n$ } exists and equal .

Proof. Assume that and define a new sequence 2 v( J$ f! L W! v as follows:

1 K3 H4 Z/ K8 V7 C7 V0 Q3 h3 H9 K (8)

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since & k2 A( d0 y2 h, [" k& ~ . We will prove next that converges uniformly on 2 {' G$ V, X$ r3 p . If , we have

, (9)

+ V* l6 ~9 ?1 O9 P

where , I& w- y9 J9 a! D! p: P . Now 9 \% A: }- Y# @6 _# { exists for each x in and has the value , F, {- u& H! b6 s7 Z! v . Applying the Mean-Value Theorem in (9), we get

, : \, R- ]0 z, p! z' ^6 K (10)

where $ {3 Q5 U# s8 A6 { lies between x and c. Since converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on 2 |8 a9 T" ]5 t W+ S .

Now we can show that converges uniformly on . Let us form the particular sequence corresponding to the special point ' x/ W [( e: k: Y) t6 I for which is assumed to converge. Form (8) we can write

an equation which holds for every x in . p0 o. j# r% H J# C$ M . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on % n4 Y( ~9 L) i . This proves (a).

To prove (b), return to the sequence # B2 d' z- m( @ defined by (8) for an arbitrary point c in 1 U7 U) o4 O6 { and let 4 i' W4 ^0 I0 T/ e . The hypothesis that " V7 g7 N# B ^1 b. x- X exists means that . In other words, each 7 S) h' j% v5 u; \ f( X/ d) j2 M; W) Z' \ is continuous at c. Since w/ [5 T: d; l+ T 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 6 X q( C3 C1 D, V+ G7 T) _! C , we have

Hence, (11) states that the derivative exists and equals . But

hence P/ p! a( u0 J7 t, w . Since c is an arbitrary point of 4 T- f; H' Y# f. V- s3 c , this proves (b).

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

Theorem 9.14. Assume that each 9 U7 C' j+ c J! _2 v) X is a real-valued function defined on such that the derivative ! i' B- ?$ H7 k9 ^9 z# I exists for each x in ) F. y+ w6 @, |8 T f) W* z . Assume that, for at least one point in , the series converges. Assume further that there exists a function g such that (uniformly on ). Then:

a) There exists a function f such that (uniformly on + \1 L4 i( m' l: c- v9 B% ~; k ).

b) If , the derivative 8 |, [ Q7 \6 b9 u6 a" A: a exists and equals .

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。。。。。