[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 0 N- \0 ?' }) m+ ^7 V) @3 U . Assume that for at least one point in the sequence converges. Assume further that there exists a function g such that ' s- r* y6 A+ a; e3 W, F6 t uniformly on 5 a( ~! h) N: z& R; F; } . Then:

) ?+ `7 P: N0 p' A

a) There exists a function f such that * I; {1 X4 S9 l+ q4 \ uniformly on .

b) For each x in the derivative exists and equal & t) D6 {& C# L9 a .

Proof. Assume that ! O+ T( {0 A6 U7 x/ ]5 c( v# n and define a new sequence ) g7 s- _ k: p5 k/ e! } as follows:

(8)

" m( n7 a) a8 `) @6 o) |

The sequence 8 ~, _$ s$ l+ o3 M6 D$ d" _8 O# J so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that converges uniformly on + G" D# I5 E7 r9 o N, N0 w . If , we have

+ @/ n! \0 n* Q; Z

, : [. s( Z/ ]2 W2 c) b# U* e+ _ (9)

3 \$ [$ g I) P; \4 i

where j# q6 V' B3 ^/ }( `; l* q . Now , `+ u$ h. D# C1 H/ | exists for each x in and has the value . Applying the Mean-Value Theorem in (9), we get

, ; }; l- f" K ]; ~ (10)

where ( R1 d5 S; a4 r3 T" f- C! q$ R) V lies between x and c. Since converges uniformly on 7 f3 I& V: k% P- O1 O (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on , C. E1 h6 Z+ z7 @% ]8 u( { .

Now we can show that converges uniformly on . Let us form the particular sequence , x7 z; t/ ^" J/ d Y) j: k corresponding to the special point ' o9 l8 M9 N4 a$ m0 J for which ' X; l& q$ u* M+ Z is assumed to converge. Form (8) we can write

an equation which holds for every x in ; _: A* f2 r7 L/ Z! v6 p . 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 : z' `$ H0 t7 ^) x/ b defined by (8) for an arbitrary point c in and let . The hypothesis that + y$ @6 J# z* n4 w \5 q2 F exists means that . In other words, each is continuous at c. Since uniformly on 9 D$ |; i* l; S8 c! T! M' d M , the limit function G is also continuous at c. This means that

/ K% Q! ^7 ]# ?2 i7 X (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 v4 ^: r2 g! x( Z- R . Since c is an arbitrary point of 7 }4 L! L4 M$ q6 ]7 w; T , this proves (b).

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

Theorem 9.14. Assume that each 0 \( s3 a! U. D$ Y0 f$ k- p0 I is a real-valued function defined on & O; L0 I- |3 J7 D; L such that the derivative % j. d( ^7 ]& @6 g B4 m5 U [ exists for each x in . Assume that, for at least one point 0 k* v/ v( {+ P: V9 J. U$ ^ in ( P5 @+ y7 y4 H5 m , the series converges. Assume further that there exists a function g such that (uniformly on ). Then:

a) 0 k& Z6 w; M/ g) A! E9 b There exists a function f such that (uniformly on ).

b) If , the derivative " I( c; t& s/ m9 V$ ^6 C 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？
1 b g# K ~7 p! ~8 G

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。