[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 in # v! G& {1 S3 P, G2 u/ M the sequence converges. Assume further that there exists a function g such that uniformly on . Then:

! S% J8 E6 W1 N/ N/ V/ s' i8 x

a) There exists a function f such that + y+ R0 ^; \+ Y uniformly on .

b) For each x in V( c% K: C t3 z3 E; \ the derivative 2 E! F8 O+ ]8 P6 J+ C n exists and equal 5 a( e" W8 Z2 _4 Y' o# n .

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

- @0 [$ @. F% g6 |$ z (8)

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since & x- \) Z, s! p* D' _' I+ Y . We will prove next that converges uniformly on . If , we have

& n& c# b) M3 W; s$ x9 R/ X

, (9)

/ W K: M& W; t0 {$ J3 x

where . J2 p: Q/ N, h, A# }% h6 ]% L . Now + [% ^* C0 U/ ?2 R+ g exists for each x in , K3 E/ }$ ~ v/ I9 t2 F: X and has the value 6 s( u6 Q4 V+ w v . Applying the Mean-Value Theorem in (9), we get

, ; c5 G; O. [' ^9 P- {# V& j! A" W2 G3 h (10)

where lies between x and c. Since converges uniformly on & E/ n& P+ Z0 P' O (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that e$ V9 C Y& B* k8 O# _" O converges uniformly on .

Now we can show that 3 X- b" R% O' L converges uniformly on # e8 l1 X! g9 D+ m0 b; e . Let us form the particular sequence : N' L6 {, M0 D* H9 _& B& Z corresponding to the special point ) M6 O) ]/ b8 D8 U O* f! z2 b for which ( K5 v0 ]* t, M" A2 I is assumed to converge. Form (8) we can write

an equation which holds for every x in & c- t7 f0 Q! V3 ]( P2 A . 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 - e! _4 D4 t- m& A defined by (8) for an arbitrary point c in and let ; B( K7 }8 Y. h" B . The hypothesis that exists means that . In other words, each is continuous at c. Since , T) k! Z7 O! p2 i& t) Z+ @ 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 , we have

Hence, (11) states that the derivative exists and equals 7 g0 E; N0 g& z9 B5 N/ _$ m . But

hence . Since c is an arbitrary point of ' `9 u$ m% ~1 g2 ~4 |2 r , this proves (b).

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

4 X' e$ T/ _7 ?6 o0 v2 C$ N

Theorem 9.14. Assume that each is a real-valued function defined on such that the derivative + J# r0 p9 d. f1 K' P exists for each x in $ n% o) b6 g9 @! n: D; X" x7 H" H2 i6 c3 H . Assume that, for at least one point in 7 i: p: ~# u/ I2 W" g , the series converges. Assume further that there exists a function g such that (uniformly on % M( c' S7 m! p/ O ). Then:

a) + D6 \) d1 X" U' m3 R/ {$ y! u There exists a function f such that (uniformly on ).

b) If , the derivative # C# r8 e& v, V) ^ Z9 ?3 e 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。。。。。