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标题: [求助]谁能帮我翻译一下这篇文章?~ [打印本页]
作者: angel_lys 时间: 2007-11-23 22:16
标题: [求助]谁能帮我翻译一下这篇文章?~
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% K4 k0 S* Z% z# i9 J/ r
. Assume that for at least one point2 n! F" r7 O- U; O3 |; S2 R
in3 u$ T K$ E5 A" g
the sequence converges. Assume further that there exists a function g such that4 h- G' K/ w$ M+ T6 P5 \5 Z0 N3 L4 k9 w" E
uniformly on. v8 X6 r* ?- C5 t
. Then:
8 r9 N/ ~# @" R) S
a) There exists a function f such that
# d4 [2 G8 \: h7 a. u uniformly on, [" J- s6 h+ i/ x
.
b) For each x in
9 E0 H2 s' L- _5 [! A& n the derivative" p! _# t* z) @ G) T8 F: z
exists and equal
+ W) t0 F: s; _$ \6 E. y G .
Proof. Assume that e& Y8 s2 i1 }
and define a new sequence
" W" n6 R4 f9 @* D) c! m* e; X as follows:
8 V' \3 {9 [/ E- F% m
& D* `# i' A* K0 F7 @ (8)
. j7 B" G4 H0 Z9 w& q) w
The sequence+ P5 V9 s L' h
so formed depends on the choice of c. Convergence of follows from the hypothesis, since
) n0 d/ l2 u4 V . We will prove next that
9 _% Z0 j6 H" K3 F" Z converges uniformly on* G7 F* g) ?( Q) r$ I( y8 G; t
. If , we have
& y# L6 [9 k. v+ ~
,
& \* z5 _; b1 V( | (9)
5 s- |5 T, W1 M4 P! w% Y. V
where& x, Z& w! F G( o
. Now
$ d+ k/ Z3 T) q9 |5 q# T) a1 [: V7 x exists for each x in7 b+ \8 [0 E6 @) w) l
and has the value
9 F& \! p4 W/ r1 Z& y$ B# T( A: {% V . Applying the Mean-Value Theorem in (9), we get
,
' n+ n- d: C/ \6 U& L; d . ?8 c9 M& g4 z$ d, h$ g9 u
(10)
where
: c. Z3 _" d4 Y lies between x and c. Since3 `) Y; k/ b3 C x$ `
converges uniformly on
! U1 x% j* @4 G (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that$ R! y8 q, b+ k6 o) p; d
converges uniformly on' o5 e K/ H3 j/ H2 m
.
Now we can show that4 T" J1 B. J, x3 k6 q2 N
converges uniformly on; N( c; ]- J) Q% A
. Let us form the particular sequence# U$ A% ?- @ [% u
corresponding to the special point- K/ f' m. D* ^
for which
4 y G3 Q! o" _2 v7 j; T0 x is assumed to converge. Form (8) we can write
an equation which holds for every x in
4 j$ N; s+ t3 \8 ^ . Hence we have
This equation, with the help of the Cauthy condition, establishes the uniform convergence of on% ~- {3 g: {9 M6 @& [7 T% ?. I
. This proves (a).
To prove (b), return to the sequence8 h; u$ U7 S+ `% L
defined by (8) for an arbitrary point c in: V7 e( m/ U" d0 J2 j
and let
9 P f1 H0 Z: o* m) X- i . The hypothesis that
3 j% E4 X% H; k- i4 X; }# l exists means that . In other words, each$ y6 `) J' w3 |2 z' Y
is continuous at c. Since
$ E' Y" {& d8 R% _: e, N% o$ A% d D uniformly on
5 P6 E7 ^- S/ x2 z , the limit function G is also continuous at c. This means that
( B9 O+ m6 K' l; I
(11)
the existence of the limit being part of the conclusion. But for. O# A/ w" A4 x) A9 k) F9 X
, we have
Hence, (11) states that the derivative1 B8 V y" P% `# X: B& M
exists and equals8 b, `, W" u0 M+ \" G. n, ^9 w
. But
hence8 K @, n8 A; U5 ]2 K* ^
. Since c is an arbitrary point of( C* j: t6 D4 |3 O
, this proves (b).
When we reformulate Theorem 9.13 in terms of series, we obtain
/ h( y R( k7 G6 N, b, g' k
Theorem 9.14. Assume that each
) w5 |8 c# m4 l. A+ ?, V is a real-valued function defined on/ F: h* k& W4 {; u
such that the derivative
" h2 e' t( L8 ~ exists for each x in1 b- m+ ~8 y3 A5 I6 d. U
. Assume that, for at least one point* p2 M8 c* j0 ?5 l. y
in. k' r' Y* b5 O! q8 u1 ^2 s. b
, the series2 T0 ~8 o4 x) _5 F* M
converges. Assume further that there exists a function g such that (uniformly on( v! M% ]* i" f+ l) I4 L
). Then:
a)7 p* v4 o- R- a, U K5 Z
There exists a function f such that
" H) `' H" K5 \/ C (uniformly on( f6 h9 h; A1 A/ X% m
).
b)3 i( s4 B8 w5 `3 { [' I4 q: d
If , the derivative! D; |6 Y M; w& N( k
exists and equals5 y7 ?& ^, ^* Z& j0 ^
.
作者: lzh0601 时间: 2008-4-19 13:22
我怎么只看到脚本文件呢？
作者: tinkertinker 时间: 2008-7-28 14:22
“Theorem 9.13. Assume that each term of is a。。。。。” 这里面有没有漏了字？
7 R/ _2 W7 W9 h/ R[此贴子已经被作者于2008-7-28 14:24:38编辑过]
作者: rao 时间: 2009-1-13 21:47
看不清楚。。。。。
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