[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 2 T1 h, `5 W* D0 Y: t& d0 K/ x# _ . Assume that for at least one point in ! P) ]6 A$ f( [' L the sequence converges. Assume further that there exists a function g such that uniformly on . Then:

7 b) O: h+ x$ \# |

a) There exists a function f such that uniformly on .

b) For each x in the derivative exists and equal ' C) A. w5 i. m: k .

Proof. Assume that and define a new sequence as follows:

$ v8 T" I7 S: P5 R

$ J$ ?2 G A; Y0 x! H2 ^ (8)

" Q: k1 T9 f; q' u

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that * G6 Q8 l& i, Y9 ^% U2 x converges uniformly on " Y) v8 l5 C3 ~0 y; ^7 S, G b . If , we have

, 3 q% ?9 g5 d4 ^+ s (9)

where 7 f$ _# r: I( s% [6 K1 C9 N . Now exists for each x in and has the value 9 D, e! W" W( H4 H% z0 |* @ . Applying the Mean-Value Theorem in (9), we get

, (10)

where ! |' @5 {3 v% n 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 .

Now we can show that . v! b: O+ {) ?3 v7 v3 w, T) N converges uniformly on ' w: W( x1 X- o3 w . Let us form the particular sequence corresponding to the special point ! I j6 ?7 z9 g- j 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 defined by (8) for an arbitrary point c in # Z) ^" T) ~, v r) N and let ! B2 m6 K; X8 f& k' Z# [$ \6 V3 @; @ . The hypothesis that 1 J% E: H6 Z2 @: h9 ~" ~; [5 d exists means that . In other words, each 9 x4 L# t6 d, g ]9 W' H is continuous at c. Since uniformly on 6 O7 i4 u' F) y# A ~ , the limit function G is also continuous at c. This means that

5 O% D% z( ^. p (11)

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

Hence, (11) states that the derivative 1 c7 D" D* S3 K2 W1 t exists and equals . But

hence . Since c is an arbitrary point of $ m, D/ S2 n7 _; z4 S+ I , this proves (b).

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

7 c7 Q" K. Q' r

Theorem 9.14. Assume that each ( u2 Q( d3 H! @) n" F: Y is a real-valued function defined on - K. ?% l- T) r such that the derivative 2 b1 I) P2 t0 }6 [ exists for each x in . Assume that, for at least one point 5 b3 B- p% X4 a4 o' H in ! g! W) i0 c: p+ Y Y , the series converges. Assume further that there exists a function g such that (uniformly on ; m; O& g9 t, N2 G) _7 a ). Then:

a) . S o" a8 L9 k; c There exists a function f such that (uniformly on 8 l9 _% c0 z$ h0 F9 T4 k ).

b) ) d2 U0 U: }4 p+ j If , the derivative . e! c+ c- Z9 d( P; U exists and equals 2 I' W+ B o. j3 D .

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？
" k v5 U8 q, Q4 G: u4 Z$ ]

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。