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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* @8 I- Z d! P* K% v1 E6 o) n- k7 F5 \ . Assume that for at least one point L% @1 O% }, h C in ) L2 q: A) ?7 S* F: A) l the sequence converges. Assume further that there exists a function g such that 4 z) \; y# Q( [! M5 } uniformly on " g: }4 f8 a: I$ A9 w" A . Then:

1 D/ E3 K8 ~+ |$ a: c2 _

a) There exists a function f such that + c3 w8 ?8 Y( v/ i! t4 i uniformly on) A5 {4 e* D+ D .

b) For each x in! k. U% [5 ]" {" _* e6 W4 q+ L the derivative & ~. ^# M' e* ` exists and equal+ w" e1 p6 Q! X, k" D .

Proof. Assume that9 l0 f0 n8 C6 D. X* Y! @( S and define a new sequence 2 Q, }' n8 l* t! `) e% V/ n3 ^( L" D* V as follows:

, z) _- d' v& Q& I3 J; t8 A$ z7 N8 e

+ |4 a$ L8 s4 |& a8 G+ O4 W- D" E (8)

4 c4 L. h% o! n

The sequence3 d+ H5 U+ f" `2 h5 g so formed depends on the choice of c. Convergence of follows from the hypothesis, since . a. t# B! t5 S8 j' ]" O . We will prove next that( S: T" J( o% [5 y converges uniformly on7 t: S% a' b$ u+ M . If , we have

2 w; ?7 l3 |- E7 \% i9 E

, ( Z" c/ M& Q" c* {& j2 a; h# F. ~! @$ \ (9)

8 R) j- l! { E2 o3 ]' R

where 5 M: b7 i, \ p1 f3 P7 Z . Now6 z: p8 W2 ^$ x, R exists for each x in7 X) D u# N5 U7 C and has the value ) e7 p2 A/ I6 U5 |" Z . Applying the Mean-Value Theorem in (9), we get

,' A+ ?% o; H( R7 m; t! Y9 y6 I " ^: X/ a" l* N3 [' C+ W (10)

where0 w [# ]" X% I" H, u: `% D lies between x and c. Since* f- D A: B) |) t0 ~/ k1 | converges uniformly on. W* O2 F% j% t. Z (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that & A: u9 C) l" N# g3 q converges uniformly on* N# ^2 [! \' f6 X .

Now we can show that: G( U$ T6 ~; k converges uniformly on / R0 s; S& x$ A6 h( d W- K . Let us form the particular sequence1 ~" `2 G* q6 `# V- t P* s! y8 i0 Q corresponding to the special point # H+ ]; I$ P; \: J+ j for which ' \) X/ D6 b$ C9 K- s4 L is assumed to converge. Form (8) we can write

an equation which holds for every x in7 ~2 H9 [) q' z+ K/ J . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on; Z5 ]. \4 G9 B) O+ T . This proves (a).

To prove (b), return to the sequence% a4 s8 y$ |+ H! j+ H4 C$ d6 b defined by (8) for an arbitrary point c in- s4 i2 h" u$ Q3 i4 j+ f$ x1 w1 }7 d+ D and let. s# K. E6 N2 K/ S' j . The hypothesis that- b! R) z4 d" h5 |$ w exists means that . In other words, each * _+ E- C0 \- i6 A is continuous at c. Since. I. _9 a7 D& n& f" Y0 T5 t, o. k3 p uniformly on( F& z9 i; }& r( n8 X , the limit function G is also continuous at c. This means that

% u; r3 n, r6 L1 x+ A8 m% v (11)

the existence of the limit being part of the conclusion. But for y- u( N& H C' f- M1 J1 U , we have

Hence, (11) states that the derivative * i# E6 \- R8 W; }& }2 @9 O4 f, | exists and equals7 }6 ]: X# W, L . But

hence 0 J: `6 X) P% v% D . Since c is an arbitrary point of ; ? R8 _3 C, n4 o q; K , this proves (b).

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

: B5 e0 V4 i. U& m- [2 v

Theorem 9.14. Assume that each " J; o3 q, Q% }( J is a real-valued function defined on3 q9 T: }2 V5 F such that the derivative " C7 Q5 |( S2 W$ S! w: _ exists for each x in# Q" u' g, r- } . Assume that, for at least one point* e, i( _9 H& p" W in9 t+ G7 S0 ]. C , the series1 R* P( i8 p& L- W; E5 A converges. Assume further that there exists a function g such that (uniformly on- r( \" M9 q) E6 N- V. P ). Then: