[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 6 ]3 f1 l. n4 E, m . Assume that for at least one point * E( P8 i8 z: R3 {3 W5 u in # [4 W+ ?* e9 {2 u the sequence converges. Assume further that there exists a function g such that 3 t" M: o) K5 z! M" N0 n uniformly on . Then:

a) There exists a function f such that 3 I7 `6 `( z2 z9 P uniformly on 1 @. u8 r* k( S% m .

b) For each x in the derivative 9 ?" D+ j" ^4 d/ H exists and equal 6 J& O8 r A! A& H7 j4 W) L( ] .

Proof. Assume that 8 _: A. |4 Z8 u4 ] and define a new sequence ) W" @ k3 y; o7 M1 d as follows:

* v% P+ G% K3 v/ ]) H

(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 + q& d2 f5 G8 j( S . If , we have

- O# g/ I+ S0 p" N& Z# t

, (9)

where . Now " M9 c. V; o3 _8 K) ^7 z& E) M9 m exists for each x in 3 p* r2 w! `" _' m and has the value - }7 u1 W, m' ?: |. b4 _" L . Applying the Mean-Value Theorem in (9), we get

, % [3 u3 s5 T6 V9 V1 ~) N (10)

where % H3 s+ T; d9 j' c' c) i, V lies between x and c. Since converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that 5 t! e" \" B& j0 D! R. H7 K converges uniformly on .

Now we can show that converges uniformly on . Let us form the particular sequence * Y5 W+ X6 O \9 k) |# L corresponding to the special point $ M! d, B1 x6 x% R. W for which , R9 H8 f9 n! q/ q; S is assumed to converge. Form (8) we can write

an equation which holds for every x in 7 T9 ^: g7 q/ V, L . 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 : j5 ]8 |) `% f' w% M! V; k and let + D! z1 K, x! Y . The hypothesis that 1 Q+ e/ p/ }9 \2 H" E, X exists means that . In other words, each 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 1 d+ @9 Y& G1 Z4 ? , we have

Hence, (11) states that the derivative 0 S- ~% q" f+ r exists and equals . But

hence . 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 4 _" E4 Q* g% B6 F v3 g such that the derivative 7 L" B! v- b0 ]. Q* m exists for each x in . Assume that, for at least one point in 1 A4 x, D* T; h7 S8 F' }9 H , 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 0 k4 e3 ]# Z" |4 c F$ Z2 F3 A ).

b) If , the derivative ) H4 ~+ F$ G7 _. H( B+ t exists and equals ; f, y4 T8 Y2 q+ e .

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。。。。。