[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 : }' z$ q* u( [3 w0 e . Assume that for at least one point . y7 w8 e9 b2 u7 h) |% R& F in 1 H7 x" i6 p0 j- L7 Y3 E7 B the sequence converges. Assume further that there exists a function g such that 6 l! Y h j) h- E7 E. h uniformly on ( H6 S, O3 e8 u3 i1 ~* h9 C . Then:

+ e( E z2 F7 Z

a) There exists a function f such that 4 N& B7 M, f- ^, O+ _8 q! f* ^ uniformly on .

b) For each x in 0 b, l. \3 z. I. D8 Q the derivative exists and equal .

Proof. Assume that and define a new sequence 1 w1 |2 f$ Q- q as follows:

& U) H/ Q9 q/ f) S y9 q (8)

3 A1 w. R7 k5 h$ m& G

The sequence 9 u* B7 a0 K% t; E9 K9 h" L- X so formed depends on the choice of c. Convergence of follows from the hypothesis, since & E$ r: N% J( y+ ~0 a0 G . We will prove next that converges uniformly on 2 s2 e' ^" ?' E7 [3 N! q . If , we have

, + u/ Q2 T/ Q" D/ T (9)

where . Now & A }$ ^% ?- Q6 G# v8 F( o6 u+ g exists for each x in and has the value % ~; m6 x$ a7 ~ N . Applying the Mean-Value Theorem in (9), we get

, (10)

where lies between x and c. Since . b; E4 p5 J; S4 B converges uniformly on 3 q' W! P9 x9 q6 e2 H (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on : g: }! G* j4 F4 Q .

Now we can show that converges uniformly on 9 V: I2 L4 J, e% P, } . Let us form the particular sequence ; [2 c) N/ R J: L/ I" ~$ b corresponding to the special point 2 K3 m* Q: x4 l5 u' r" d9 f 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 ( N( v. [6 a, i8 v. w9 t9 } . This proves (a).

To prove (b), return to the sequence " q* p' F6 i Z" ?* S2 K) B defined by (8) for an arbitrary point c in and let . The hypothesis that 9 t$ `+ C5 B# Y! t exists means that . In other words, each is continuous at c. Since uniformly on / L; Y5 R. Z) q5 e , the limit function G is also continuous at c. This means that

% \( {5 S7 I# ?! ], }" B% o4 A (11)

the existence of the limit being part of the conclusion. But for ' e$ g8 i- e; Q0 N8 }, z5 ` , we have

Hence, (11) states that the derivative 9 {8 S* w1 h9 q exists and equals N1 K# ~6 E/ ~9 F . But

hence ~4 N' W& I% i" `( U4 B' O . Since c is an arbitrary point of 6 N8 F1 g# V# F: h3 y! h6 s, i6 X , this proves (b).

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

& D# A+ \9 h+ w' m

Theorem 9.14. Assume that each is a real-valued function defined on such that the derivative exists for each x in . Assume that, for at least one point in , the series # ]8 _. u6 a. U3 @) m converges. Assume further that there exists a function g such that (uniformly on * X/ E( Y: H( z1 N7 N- n ). Then:

a) There exists a function f such that (uniformly on 5 V0 c e( h' j ).

b) If , the derivative $ m" K9 `" D7 [2 X& }& A exists and equals / e' C% L/ q1 j0 Y .

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。。。。。