s****r
发帖数: 47 | 1 L2里的函数都满足PARSEVAL定理.如果一个可测函数不在L2里面,它的能量是无穷大,那
是不是意味着它的傅里叶系数的平方和也是无穷大呢?无论是不是,有什么办法证明吗? |
G******i
发帖数: 163 | 2 If f is in L^1(0,1) and if f 的傅里叶系数的平方和 is finite,
then f is in L^2(0,1).
Proof.
Let f_N= the N-th partial sum of (f 的傅里叶series).
Since f 的傅里叶系数的平方和 is finite,
f_N is a Cauchy sequence in L^2 and hence converges to some g in L^2 norm.
Easy to see (g 的傅里叶series) =(f 的傅里叶series).
Thus, g=f a.e. |
s****r
发帖数: 47 | 3
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~为什么FOURIER系数一样两个函数就一样?g在L2里,但f可不一定在啊.
【在 G******i 的大作中提到】
: If f is in L^1(0,1) and if f 的傅里叶系数的平方和 is finite,
: then f is in L^2(0,1).
: Proof.
: Let f_N= the N-th partial sum of (f 的傅里叶series).
: Since f 的傅里叶系数的平方和 is finite,
: f_N is a Cauchy sequence in L^2 and hence converges to some g in L^2 norm.
: Easy to see (g 的傅里叶series) =(f 的傅里叶series).
: Thus, g=f a.e.
|
B********e
发帖数: 10014 | 4 同问
really a subtle question
啊.
【在 s****r 的大作中提到】
:
: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
: ~~~~~~~~~~~~~~~~为什么FOURIER系数一样两个函数就一样?g在L2里,但f可不一定在啊.
|
G******i
发帖数: 163 | 5 That's a basic result about Fourier series.
If the Fourier series of an L^1 function is 0, then the function is 0 a.e.
啊.
【在 s****r 的大作中提到】
:
: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
: ~~~~~~~~~~~~~~~~为什么FOURIER系数一样两个函数就一样?g在L2里,但f可不一定在啊.
|
B********e
发帖数: 10014 | 6 if i'm right
1. the result is about L^2 function
2. no matter L^1 or L^2, the result works only if both of the functions belo
ngs to it, isn't it?
【在 G******i 的大作中提到】
: That's a basic result about Fourier series.
: If the Fourier series of an L^1 function is 0, then the function is 0 a.e.
:
: 啊.
|
G******i
发帖数: 163 | 7 ???
Sorry. I don't understand what you are asking about.
Do you mean that the theorem I cited is incorrect? |
B********e
发帖数: 10014 | 8 呵呵，nononono
我记得这个定理对L^2的,只是想确认一下对L^1是否也对，希望能得到
个说明或引用出处；）
if it's right in L^1,certainly you are right because L^2(0,1)\in L^1.
【在 G******i 的大作中提到】
: ???
: Sorry. I don't understand what you are asking about.
: Do you mean that the theorem I cited is incorrect?
|
B********e
发帖数: 10014 | 9 双修侠，忽然想起还没有得到你的确认呢
是不是对L^1也成立啊？
谢了
【在 B********e 的大作中提到】
: 呵呵，nononono
: 我记得这个定理对L^2的,只是想确认一下对L^1是否也对，希望能得到
: 个说明或引用出处；）
: if it's right in L^1,certainly you are right because L^2(0,1)\in L^1.
|
G******i
发帖数: 163 | 10 Well, I stated the result accurately & precisely.
I can't add anything to it.
You can find it in any standard text about Fourier series.
【在 B********e 的大作中提到】
: 双修侠，忽然想起还没有得到你的确认呢
: 是不是对L^1也成立啊？
: 谢了
|
B********e
发帖数: 10014 | 11 可以随便指一本给俺吗？
【在 G******i 的大作中提到】
: Well, I stated the result accurately & precisely.
: I can't add anything to it.
: You can find it in any standard text about Fourier series.
|
c****n
发帖数: 2031 | 12 An introduction to harmonic analysis, by Yitzhak Katznelson
【在 B********e 的大作中提到】
: 可以随便指一本给俺吗？
|
B********e
发帖数: 10014 | 13 really a nice book,thank you!
【在 c****n 的大作中提到】
: An introduction to harmonic analysis, by Yitzhak Katznelson
|
