[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 " k6 ~/ p. b; k. L0 i9 `* V in ; O& B, {$ z, L1 I. e the sequence converges. Assume further that there exists a function g such that uniformly on . Then:

/ q1 ~5 Q0 x5 Y! x+ `2 G

a) There exists a function f such that * N+ p. L. _& M M3 Q uniformly on / b6 K" Z6 ` E" `0 Y5 \ .

b) For each x in 1 h2 o# } G. t the derivative exists and equal .

Proof. Assume that " F/ p/ z! a0 O and define a new sequence ' \3 ]# E# r$ x! _ H8 z' U( } as follows:

. I _/ g: Q1 b" o6 r7 [

: S% J& g# d# p# k) q (8)

5 Y1 [5 U: | c* T

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 . If , we have

4 v' W/ u& D% t3 e

, (9)

where 9 |) v: w2 Q$ C2 f$ x# f . Now % ]" u0 `" R, e$ T# ~ exists for each x in and has the value . Applying the Mean-Value Theorem in (9), we get

, * W, _( P1 e T2 t" t. C, r( m# w4 a$ C (10)

where 4 X5 ]5 M8 C! A/ H lies between x and c. Since converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that . r/ u3 H1 W' E D1 b converges uniformly on .

Now we can show that " X; i5 y- k6 o3 _/ T2 @6 ]! K converges uniformly on : L. R7 V7 P' B& Y$ W4 m . Let us form the particular sequence / r) a9 g: |/ X% G 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 . This proves (a).

To prove (b), return to the sequence : W6 m/ {& H% s+ }4 s; H7 O defined by (8) for an arbitrary point c in 0 o4 f! \% {0 i$ _3 E and let . The hypothesis that exists means that . In other words, each - E8 t( e; k+ O& h is continuous at c. Since 2 p( ^( ^! v0 m+ l3 N uniformly on : h6 p- C3 w" J% @8 e+ G , the limit function G is also continuous at c. This means that

/ g# r/ P1 n" e" j (11)

the existence of the limit being part of the conclusion. But for , we have

Hence, (11) states that the derivative exists and equals + g( n) Y; y; a& s" W . But

hence 9 M6 y: E1 ^, y6 I, M; } . Since c is an arbitrary point of / }5 q& \( T$ k , this proves (b).

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

8 d! Q- n8 v4 i& q

Theorem 9.14. Assume that each is a real-valued function defined on 6 F# u! r* C5 [8 T. Y3 E# v such that the derivative 4 ^1 U# c4 c1 G: S8 k exists for each x in . Assume that, for at least one point & Y6 f% y' M. _2 d in , the series converges. Assume further that there exists a function g such that (uniformly on 9 l2 w" I+ a) s7 o) Q6 N% Z) C ). Then:

a) - a! |1 p7 B" H. t0 ^9 U3 K There exists a function f such that (uniformly on ).

b) ' C% W- }1 }- l7 F If , the derivative * D# }. ]+ F% O' {. x, S exists and equals , O0 P" n" M' O; l! W) f .

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？

[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？