[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 # p3 `; k! c0 x% K/ d2 X7 ^/ V& N in the sequence converges. Assume further that there exists a function g such that # x; ?3 E, [& n1 } uniformly on ) u. x8 e5 F% ` T& q . Then:

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

b) For each x in 5 @7 l3 d& X9 O6 B3 w1 { the derivative t8 l* |0 X) {- z exists and equal 4 b4 ~ ^$ f5 s$ R/ o .

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

4 F( L$ y R5 p5 `9 M/ _# F: c/ @ (8)

2 D5 Z4 w" p9 Y1 D3 [

The sequence . X/ i+ Y* u) ^* k& t2 I so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that 2 Y9 H1 d) d7 f0 k( X! s: ^; C converges uniformly on . If , we have

, (9)

|/ m' U6 H( E2 E* r

where . Now exists for each x in and has the value & U. h& l- F1 `. `: h . Applying the Mean-Value Theorem in (9), we get

, & l& f: `& L+ [1 w) I * } z3 \/ t+ D# |2 T+ M3 @+ `# j (10)

where lies between x and c. Since ) m: s( G Q1 |. o+ H2 q 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 ( ?/ o* B, T+ {& G% I3 |4 g converges uniformly on ; z0 i4 N. w( ]. d5 }' b2 @ . Let us form the particular sequence corresponding to the special point for which # f2 o9 e$ D7 i/ a% z7 ` 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 : F' [: v/ E$ F6 Q . This proves (a).

To prove (b), return to the sequence 8 v2 w2 M2 z4 }" }, o( l/ J defined by (8) for an arbitrary point c in and let & D) V) h3 }) t* @& e2 P . The hypothesis that 6 \, Y( Z3 u5 E u; U exists means that . In other words, each 2 K; l" o! r9 G: c3 K, M9 C is continuous at c. Since 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 I" L$ B6 o# W% x# G4 q , we have

Hence, (11) states that the derivative exists and equals . But

hence . Since c is an arbitrary point of , this proves (b).

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

Theorem 9.14. Assume that each 3 \% s2 ]& t5 a is a real-valued function defined on such that the derivative exists for each x in / D. p; z& ?* M1 Z" N) A . Assume that, for at least one point in % O. |, o1 \$ s; p( J , the series 4 F2 e) s0 n- U4 M converges. Assume further that there exists a function g such that (uniformly on ! z u$ n2 T5 [: W" j0 V ). Then:

a) 7 r, f& q; ]! |) L: }1 E) _ There exists a function f such that ' i' I4 q. ~( p: X5 E4 q% f (uniformly on ).

b) & }2 T2 e2 W8 L( [3 P) M. L If , the derivative exists and equals f6 ~+ t. O/ i: F/ ^ .

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。。。。。