[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 " c/ O7 G( D/ n8 s( _% F$ p . Assume that for at least one point 6 p; A/ w8 {1 n: z. H in " Y8 l' J3 ?- e9 j" x a7 B the sequence converges. Assume further that there exists a function g such that ; r' J" d9 M% H7 H uniformly on . Then:

a) There exists a function f such that uniformly on 3 T5 s* k ?8 z# M5 W; B# P: A; V .

b) For each x in 4 w8 A8 V( ]5 h Y4 M" J, b) ]! P& c the derivative 6 D4 V1 s4 S0 A! L exists and equal .

Proof. Assume that 1 Z7 H4 ?: N% l! S7 p and define a new sequence as follows:

4 F& ]+ H0 x5 t& ~

(8)

( v4 G$ ?- j0 p- B% J

The sequence * j1 N, n, [, s$ w$ G# M& ^ so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that . f+ J% G/ O8 C' V4 t0 w* R converges uniformly on * X T( h# {8 o1 S! g . If , we have

+ o1 u% u7 R }8 s

, 7 P: r5 f3 r7 `+ N3 ~ (9)

where ; g5 r+ G3 F( r+ F+ G) o. R* n. S& s . Now 8 p1 p [6 y& k3 b8 _ exists for each x in and has the value % [. J0 ^* d: G* D9 O5 w6 |% J# u5 T . Applying the Mean-Value Theorem in (9), we get

, (10)

where 5 R2 U0 j& \: e6 a lies between x and c. Since 0 y* w* G$ Q3 o5 U! |2 J* C converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on + Q! G7 s, T% y$ I1 P- U. R2 }* c .

Now we can show that converges uniformly on 9 w( u7 q/ R9 b( _; r) Y ~ . Let us form the particular sequence corresponding to the special point for which is assumed to converge. Form (8) we can write

an equation which holds for every x in % B- I; T/ i% l e/ T1 D8 V& O . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on 9 u- u7 e9 g; ] . This proves (a).

To prove (b), return to the sequence defined by (8) for an arbitrary point c in and let + Y7 X4 W+ L, Y& T . The hypothesis that ' F+ O6 W5 i% W+ f' ?% m exists means that . In other words, each , o8 C# p0 Y8 X: x3 t) A is continuous at c. Since uniformly on , the limit function G is also continuous at c. This means that

# ]2 Q, I2 G# Y1 g (11)

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

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

hence + ~* a# x) [* [4 I4 R/ m1 O . Since c is an arbitrary point of 7 W2 t6 ?# ?/ C, d: j; x , this proves (b).

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

; l3 h8 l4 \9 _

Theorem 9.14. Assume that each is a real-valued function defined on such that the derivative exists for each x in ) I0 [7 s- p4 \* P- J2 N . Assume that, for at least one point & W6 w6 N+ Q5 Z5 Y' O in ( A! {% V: W. T' A: r , the series Q {# L$ P- c# n# x+ {* \. u% y converges. Assume further that there exists a function g such that (uniformly on ). Then:

a) There exists a function f such that 6 ]9 W* b4 p% U2 T# \ (uniformly on ).

b) 8 ?' F' s: N% T4 h% x 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。。。。。