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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; `/ [ z3 m6 i+ t . Assume that for at least one point1 Q/ T% a. K7 ]) n1 y* m" A- {# j in# h& K+ Q6 \$ f the sequence converges. Assume further that there exists a function g such that6 [0 g: _, H& B# L6 {5 N+ V, k: D uniformly on; s! J" b& B# f7 i$ a . Then:

9 b; b5 f" i; ~# F3 M, T$ A

a) There exists a function f such that / j9 {. T G1 D uniformly on / i0 Z) u. I i2 Q8 S5 r/ i .

b) For each x in " r0 z$ Z0 E3 B the derivative$ D" `8 e/ T* V/ j6 {2 _, t exists and equal' n0 p* Q% p3 M .

Proof. Assume that3 d5 H5 q- q% [- K and define a new sequence- X3 B; t1 i% o( `9 I as follows:

: c1 _+ O. x- U% o/ v# n/ z( Z

5 W: f: w P! [$ L$ a (8)

% @* f5 h4 ]% P- N0 V& x( Z

The sequence# b1 A' m# l$ x0 d so formed depends on the choice of c. Convergence of follows from the hypothesis, since* T" i/ z9 I$ u p . We will prove next that ' c+ V& t2 x! C converges uniformly on) I- r4 c$ `0 i( \1 q6 Q# e2 Z . If , we have

; t, K, X& l# _

, , [7 ~2 O: z4 } (9)

0 |) x- G% k) d6 @$ ^: F# b

where2 o# a$ I& Q& d* \6 w . Now0 [8 |4 }' B9 }8 L5 Z* X+ F exists for each x in/ Z. @% ?5 {$ d: ?6 _( | and has the value9 r( h' E. _ @/ C7 c9 r5 _ @ . Applying the Mean-Value Theorem in (9), we get

,) q2 Y5 Z/ q3 |8 F* y 6 t7 h$ Z4 w) s* } (10)

where- k6 i# X3 C# B( F: [8 M! S lies between x and c. Since 1 H6 Q" I3 T [5 W5 b: G; c. R converges uniformly on2 ~+ p f: }/ X* J/ m (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that/ C% l+ `4 T! o7 v6 } converges uniformly on8 K$ K; P+ K9 I' G .

Now we can show that9 g) N4 e- }% f( I' Z converges uniformly on , X$ z: M M' F/ l . Let us form the particular sequence: V, K) O9 T$ [: V4 c corresponding to the special point 8 t& X) @/ G ^$ [/ B8 i for which! }( ]$ Z: b" |4 m is assumed to converge. Form (8) we can write

an equation which holds for every x in & V7 D* @2 `) x . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on, @$ Y& Y0 X2 M9 Z/ Q. i . This proves (a).

To prove (b), return to the sequence1 m! d8 F N' I: ` defined by (8) for an arbitrary point c in! ~) t7 D7 ~5 k3 s+ ~4 U and let ! O: f/ b1 @2 n& Y1 X/ G . The hypothesis that ! s8 q& C* l: F7 S exists means that . In other words, each / q7 s8 F# ?0 V, p is continuous at c. Since; V1 b* ] O/ h% l, A uniformly on 7 }( B( L/ y4 I* q8 O( p5 y , the limit function G is also continuous at c. This means that

( I7 |3 A% `: ~# a! x x" p3 U (11)

the existence of the limit being part of the conclusion. But for ) L Z% k) v& f8 U3 w+ I , we have

Hence, (11) states that the derivative 9 h1 ^4 |* y9 D: G3 g) H exists and equals % h% m0 ]! k: V' m- [; D i . But

hence & `8 l1 g3 V& r- G$ K8 e- x . Since c is an arbitrary point of+ }0 w1 n; J) V+ n' G+ J* J, f7 | , this proves (b).

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

$ W# e9 t$ [6 b, `

Theorem 9.14. Assume that each: r9 Z" g* N# @0 h$ ? is a real-valued function defined on 1 T; M, L* M2 r6 N$ F) q such that the derivative5 ?! [2 Z* y- ~/ O' w, ^7 m/ { ?" Z3 X exists for each x in- v9 `5 ?( F1 |% z8 r . Assume that, for at least one point 4 W/ s3 W% ~* w- F* u" m: Q- Q in* M k* \% [" j7 V: @ , the series * H, v( |, {. L( F; J" t converges. Assume further that there exists a function g such that (uniformly on B) E, w5 d3 u/ [ ). Then: