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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! k. t3 J& j3 X! f W$ x0 d . Assume that for at least one point " c+ ]! N! y+ W+ {+ ~! M; h1 N, e$ X in- T g# ~. \: K/ K the sequence converges. Assume further that there exists a function g such that / t$ }$ I/ U7 y2 v1 R! n- F$ j uniformly on9 { R2 U( ]4 l5 s+ |$ t . Then:

0 W, d" W, r" e9 i" J+ G

a) There exists a function f such that z9 E" J3 F. V4 {+ D uniformly on ' ]- T) @8 t* T: B .

b) For each x in , L$ m2 g$ e% F M: w the derivative0 U5 ~- \1 s% E9 R: i! \2 s% ? exists and equal" H: ^" h, Y7 D% L# q .

Proof. Assume that3 q# j: `/ W4 n: B. I) c and define a new sequence 6 C/ J# S" c# J* P! m, c as follows:

& u6 \( _# m1 y5 Y" P" Z; E

& E" z) I: B6 o# g# f (8)

$ r7 Q2 ?" p, C: y& M9 k

The sequence9 N, ]/ w' c. E. C) M7 @ so formed depends on the choice of c. Convergence of follows from the hypothesis, since 3 k. o& v# I/ u4 V% n . We will prove next that L! B+ I# t* c/ `- d4 J N3 j converges uniformly on% e4 p U0 f- }' \) y . If , we have

- R) @, j) P) }& C4 a

, . Q, L: H; k8 c% q a" m1 W (9)

3 b7 F: f0 t! O/ l3 f4 J- V

where 4 H n- e5 r2 X2 H$ Y/ Y& T . Now0 M$ {7 X3 ?5 i& b' w, c exists for each x in. E1 e0 o" m- g* v6 U/ R and has the value , A3 g. [$ D' g/ s+ Q* ? . Applying the Mean-Value Theorem in (9), we get

,, ^; z1 Q* @" x 2 A, S( v0 ~; T( @; \ (10)

where; N! ?, {5 Y8 Z) H, M" z2 e8 U, ^ lies between x and c. Since0 Z" Q7 N. J: S+ @3 u8 T: W* D+ e6 A converges uniformly on ( y0 b8 i' C, Y l (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that 6 \3 }/ k% u1 s4 y converges uniformly on 0 }0 l3 A& K2 w$ A$ ?, a .

Now we can show that& L- B8 m. R( M7 F8 Q; v converges uniformly on 6 p. v e) B6 ^) ]+ d . Let us form the particular sequence \8 n# }2 t: s) x8 r4 E& y4 s corresponding to the special point * q: U0 K( y- O for which ; ^1 A! Q2 Z( e is assumed to converge. Form (8) we can write

an equation which holds for every x in5 X5 n- L* s4 ^& H! I2 i" s . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on 1 f7 a; \4 H i. F. L . This proves (a).

To prove (b), return to the sequence / h" e; ]: k' F, X: h) U0 h1 l3 C defined by (8) for an arbitrary point c in 0 G) Y; T' @' I' ^9 Y and let8 w: n, T- }; P3 ]2 [ . The hypothesis that X3 e+ ^% ]2 A. N4 d exists means that . In other words, each& F' a' f+ G) K% j+ O# ~ is continuous at c. Since & \+ r& V$ J* }0 k1 z uniformly on6 O) W0 T3 z' t+ E" u , the limit function G is also continuous at c. This means that

5 K- K7 ~( F! ?/ v2 a) ~7 ]" R (11)

the existence of the limit being part of the conclusion. But for . y0 n, g3 Z3 z. _* D9 U+ T , we have

Hence, (11) states that the derivative1 o$ @6 F) D4 p7 I" c) t* J1 h7 @5 k# V( T exists and equals- V; ~0 B9 n+ P9 B0 U3 x4 n0 B . But

hence $ r Q/ f1 y2 r5 s% \6 `( f U( S . Since c is an arbitrary point of6 W1 e* y" S# |4 s6 D2 G , this proves (b).

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

7 f' D: f& [* N: Q0 N* X1 S- @

Theorem 9.14. Assume that each 8 [% _* k* r) L X is a real-valued function defined on 8 k" H% ^1 v& E6 g0 Q such that the derivative8 u- G: G; j7 D- B/ d& q7 I- K exists for each x in ( f8 S4 h. _4 M* A . Assume that, for at least one point7 c4 ], c, `/ O; ]9 c in 7 r# M0 J& i' g/ { , the series ( l. K e/ V, e+ _2 R$ I- c converges. Assume further that there exists a function g such that (uniformly on5 ?) y" \7 P0 P' m ). Then: