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[求助]谁能帮我翻译一下这篇文章?~

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电梯直达

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发表于 2007-11-23 22:16 |只看该作者 |倒序浏览

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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 " m. [$ T$ ^8 D6 c' \% _ . Assume that for at least one point8 N% I1 L+ i! w4 \, v9 q( r in) Z( C! L9 F9 S1 _ the sequence converges. Assume further that there exists a function g such that- W4 Z: y \# R. c3 b uniformly on : H1 W" A) l5 y . Then:

0 O; w% h( n) ~, n a( `

a) There exists a function f such that . U+ _4 p( j2 i6 u uniformly on* m q: G5 F+ U3 u. {, P7 n .

b) For each x in ' {- u, m7 ^1 }' S% W1 P' ? the derivative . z" b% d) L3 E, K2 N! q exists and equal 6 A' \1 P- s* h" X: X7 j. a& `. R6 W .

Proof. Assume that - f' e! }- y9 R' M3 [ and define a new sequence % ?; Z# \& V2 N* }3 e1 H. m( f as follows:

; l" s g% R/ |5 {

& a4 u; n# ?: Z6 p (8)

0 d. [* D$ X& |; A1 V6 k* x& I1 K

The sequence$ Y) [) j& f) }! S1 \! y so formed depends on the choice of c. Convergence of follows from the hypothesis, since( m9 R9 ?# L$ c3 Z4 S0 N . We will prove next that - w! Y$ i: y* J% a converges uniformly on 0 i# P( e8 O, B& f! o . If , we have

4 P$ {0 o( p9 x/ l

,1 p" Q) ^ m6 D* O) p3 V4 I* ~ (9)

4 E, K9 Z5 a- \* t

where/ t( `8 y- C* N' d# Z . Now ! k7 e2 x& n. N$ I' x( }: w1 l+ P exists for each x in, P+ t7 h- k* t$ H) O and has the value3 i/ |; V( W. ~7 f y; d/ n* o . Applying the Mean-Value Theorem in (9), we get

,' H( @% z) ]% H% g( d" d $ w$ F9 X7 @: Q) s# O1 J2 [4 \6 o (10)

where ! s- ^! a5 [! L) U4 b) | lies between x and c. Since 7 r* }2 v$ S6 ^ converges uniformly on3 T( X. K4 H& V (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that d b/ d6 B( G. T% o+ W converges uniformly on0 R$ Y! u3 o" J; X$ o% m/ k: o4 o5 } .

Now we can show that 9 O. ^2 [: P E/ { F+ M% s/ E converges uniformly on4 W9 C, R i5 k. t0 P . Let us form the particular sequence Q, k) `, h" `$ j% h" w corresponding to the special point# W, v2 p/ r" `% ~5 @ for which f/ s5 Z( U, @. a+ E" y is assumed to converge. Form (8) we can write

an equation which holds for every x in2 n6 }! }2 f5 P5 q( K . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on) V Z$ x$ [# g K! Z. w . This proves (a).

To prove (b), return to the sequence 6 P! U) q# Z0 r9 q: X defined by (8) for an arbitrary point c in 9 [& ~& u5 a1 Z# l' i* p and let & E) H( z5 J, H6 }: c . The hypothesis that ; i3 _, G( f9 a5 D exists means that . In other words, each2 y) ? Z1 ^! P% T" o/ n is continuous at c. Since6 E1 O0 y0 Q v2 X/ ^2 A1 w- t uniformly on ( ^" F! m A( L% m , the limit function G is also continuous at c. This means that

- D9 P- D4 e+ o7 r1 N: T$ c (11)

the existence of the limit being part of the conclusion. But for, r8 p9 b$ R' `, b5 B. E+ H6 O , we have

Hence, (11) states that the derivative2 i2 k7 I8 h+ O5 c& }3 G exists and equals $ e& B- w' q0 ~: g$ r . But

hence( J; h* p7 E% F7 m! g5 C) z . Since c is an arbitrary point of Y: o# b0 j7 b# h1 z0 h" I , this proves (b).

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

9 o/ L& M( q6 N* w" d2 T3 x( i7 H

Theorem 9.14. Assume that each 5 _# }+ h8 I9 }' [% i% E is a real-valued function defined on& t% c( N0 Z) ~$ v such that the derivative% B; n# G. w/ w/ d* P exists for each x in 9 b) d% Y0 ?+ Q5 C3 A" y . Assume that, for at least one point - d1 r. j2 s p* m% r4 D) {- b in 4 A. P; k( a, M* j , the series R4 T# Y0 V& D' g6 D converges. Assume further that there exists a function g such that (uniformly on( w- z5 p" e- t5 N- P1 e ). Then: