[求助]谁能帮我翻译一下这篇文章?~

angel_lys

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 . Assume that for at least one point 1 Z& p+ Z; u: y in 3 ], Z% [4 N" r7 b4 Z# D H+ l. a: a( C the sequence converges. Assume further that there exists a function g such that ' g9 R: Z9 B* |; _. w uniformly on . Then:

a) There exists a function f such that uniformly on .

b) For each x in - c9 @4 [% v( v5 m: E the derivative exists and equal . G, j1 N, W Y1 P .

Proof. Assume that 2 ?/ E$ d) X3 ?; b2 x' J* _ and define a new sequence as follows:

9 C* H1 A, _9 P

(8)

0 r8 B k- W& A

The sequence ' s$ m/ F; N& X so formed depends on the choice of c. Convergence of follows from the hypothesis, since & C' p& @3 o" u: l* M. ^' R . We will prove next that converges uniformly on ) n4 o9 U% }6 }3 B5 U . If , we have

, (9)

8 }5 \4 u' G- r5 R

where . Now exists for each x in ( i; U5 M* ~; j and has the value . Applying the Mean-Value Theorem in (9), we get

, + s W i" v% }* k: f (10)

where / @0 U, ^' ~- R2 o: g1 R lies between x and c. Since * e0 K) M) m( o converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that ; G7 T4 `3 Q' {% T. z" P, J converges uniformly on ! E% V/ b1 m4 P8 z3 e .

Now we can show that 2 `( p) e* u* h3 d converges uniformly on - q7 R7 m. S: F/ t: W& N- | . Let us form the particular sequence corresponding to the special point for which is assumed to converge. Form (8) we can write

an equation which holds for every x in . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on ) x8 ?# a L1 C6 K . This proves (a).

To prove (b), return to the sequence defined by (8) for an arbitrary point c in ; @0 s0 x. I! X+ S, _ and let . The hypothesis that 7 _% y0 } ]1 H1 A2 _$ z4 t exists means that . In other words, each is continuous at c. Since uniformly on - T1 f0 B$ t: v p; e; M , the limit function G is also continuous at c. This means that

# e. V- K$ l; s (11)

the existence of the limit being part of the conclusion. But for , we have

Hence, (11) states that the derivative 6 y' U' K* O' p1 A4 O' W) G exists and equals 1 p+ z: U3 T; f1 ^. d) H . But

hence ; H4 _6 ]: i. [7 Q . Since c is an arbitrary point of - Y' f) Y, I( U: R , this proves (b).

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

" u, H9 a" k& I8 i# w3 C) ~/ n+ R

Theorem 9.14. Assume that each ' I. {$ `) R1 k% B* \" ~( j is a real-valued function defined on ' i1 ?$ F, f! f7 Y A4 o+ X such that the derivative exists for each x in 4 J4 S/ `0 L/ Q" Q. F+ K- B; r . Assume that, for at least one point in , the series converges. Assume further that there exists a function g such that (uniformly on . \' P# t/ |3 Y* S x ). Then:

a) There exists a function f such that (uniformly on / O" k( Y0 @# F9 A ).

b) If , the derivative - G0 U% ]9 M' V exists and equals .

lzh0601

我怎么只看到脚本文件呢？

tinkertinker

 “Theorem 9.13. Assume that each term of is a。。。。。” 这里面有没有漏了字？
. a% ^0 Y; P4 A9 r

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。