[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 H8 Q0 }# Q/ c . Assume that for at least one point # Z, b7 p# i. _/ Z7 X2 x, O+ m in the sequence converges. Assume further that there exists a function g such that 0 r9 y5 X9 ~* {8 \; u% o# F: L uniformly on . Then:

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

b) For each x in the derivative / M$ R+ _4 } ~ exists and equal 4 M' {* o" j$ b .

Proof. Assume that / D$ Q) y6 h' c7 l' i6 J, [: R and define a new sequence as follows:

1 f) h& w M, \9 e3 S/ A (8)

$ i8 L6 P R w& Y

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since 3 i! e+ R$ D# B% j8 t5 d6 c . We will prove next that converges uniformly on # Y; o/ w. F5 W- N6 k- _ . If , we have

}% X: |/ S/ i: j6 s

, (9)

! [ S. G! O; V! D" ^

where 8 z6 F0 E2 d6 W6 q- }; r5 u . Now " I5 [/ v, f$ Q2 E9 _6 Y exists for each x in * [/ s3 z( f8 J. e and has the value . Applying the Mean-Value Theorem in (9), we get

, 2 s5 O- F6 x2 W x2 s' I (10)

where " L$ o) R( g7 q. t" Y0 a$ i lies between x and c. Since converges uniformly on ) o$ h$ ^% @1 ?; T8 K4 C (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on .

Now we can show that 3 a# D) V6 N5 U/ d4 B converges uniformly on $ V! G* n j% u N: i$ l# `0 u4 A+ @ . Let us form the particular sequence ( n8 X% O q# Z corresponding to the special point for which ! q6 u$ E8 L! Z$ Y1 }3 x2 f is assumed to converge. Form (8) we can write

an equation which holds for every x in # Z) l$ m+ d# X' b8 ` . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on % \/ l$ t4 ^8 ` . This proves (a).

To prove (b), return to the sequence " J( L1 E% U3 N* T: E2 v/ P" q1 ~* Q defined by (8) for an arbitrary point c in % P: p# q9 b5 M# v3 C v" } and let . The hypothesis that A& c" A. G0 F4 @* }2 k exists means that . In other words, each is continuous at c. Since uniformly on " M+ Y5 C" K2 V: X6 L5 s" ~ , 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 ; |8 M+ O# b9 t+ W , we have

Hence, (11) states that the derivative exists and equals ) `; p& l) i. v7 e . But

hence 9 o5 P) G1 r3 W3 \6 J a# F . Since c is an arbitrary point of , this proves (b).

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

! B$ `" E# Q4 G/ D

Theorem 9.14. Assume that each ) q) e$ K; h) c$ e' C* x is a real-valued function defined on 8 E$ v F# v( l7 J- ?4 C5 R- ` such that the derivative exists for each x in . Assume that, for at least one point @; s; v5 q1 P3 }3 m% d$ B' J! L) z in 2 H {7 c9 v% E , the series converges. Assume further that there exists a function g such that (uniformly on 7 \! o, E* j/ f0 a+ X) y s ). Then:

a) / Z6 }* [# n6 V5 _9 x% D There exists a function f such that (uniformly on ).

b) \2 Q( C3 a0 E% l9 u. P2 ~. T If , the derivative 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。。。。。