B********e
发帖数: 10014 | 1 you are right,not sufficient
couldn't understand many of the theorem statements. I will try to look at
some textbooks. I did manage to find this related statement: |
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H****h
发帖数: 1037 | 2 我这样推理一下，不知道对不对。
既然F是f的FT，那么f也是F的FT。既然wF是可积的，假设g是wF的FT。
那么g恰好是f的微分。
something |
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B********e
发帖数: 10014 | 3 应该是这样的，我想多了，呵呵
distribution意义下FT的唯一性保证了这个结论
还是康牛啊 |
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j********3
发帖数: 9 | 4 Thanks again for the responses. I understand that if f(t) is differentiable, then the FT of f'(t) is iwF(w).
I am wondering if I can find a f(t) that is not differentiable yet |w||F(w)| is in L1.
It does not seem obvious to me that such f(t) does not exist?
Also, I still interested in knowing how to prove the statement in my previous post.
Thanks again for helping, I really appreciate it. |
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H****h
发帖数: 1037 | 5 我前面的忽略了太多的细节。你可以这样证明。
现在已经知道f是F的FT。把f写成积分的形式。
然后求(f(x+h)-f(x))/h当h->0时候的极限。利用控制收敛定理。
当然，如果钻牛角尖的话，你可以改动f在几个点的取值，使得f不连续，但FT不变。
differentiable
)| |
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s****r
发帖数: 47 | 6 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? |
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G******i
发帖数: 163 | 7 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. |
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B********e
发帖数: 10014 | 8 同问
really a subtle question
啊. |
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B********e
发帖数: 10014 | 9 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? |
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G******i
发帖数: 163 | 10 ???
Sorry. I don't understand what you are asking about.
Do you mean that the theorem I cited is incorrect? |
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B********e
发帖数: 10014 | 11 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. |
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B********e
发帖数: 10014 | 12 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 |
|
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c****n
发帖数: 2031 | 14 An introduction to harmonic analysis, by Yitzhak Katznelson |
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B********e
发帖数: 10014 | 15 really a nice book,thank you! |
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H****h
发帖数: 1037 | 16 You are right. Check the inverse Fourier transform.
In fact, it is also true for the FT of finite measures.
The FT of a finite measure \mu on R is
F(t)=\int e^{itx}d\mu(x) |
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H****h
发帖数: 1037 | 17 You are right. Check the inverse Fourier transform.
In fact, it is also true for the FT of finite measures.
The FT of a finite measure \mu on R is
F(t)=\int e^{itx}d\mu(x) |
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r*****f
发帖数: 247 | 18 Exponential sums as discrete fourier transform with invariant phase
functions
是Springer Berlin / Heidelberg的
x********[email protected]
谢谢 |
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g****t
发帖数: 31659 | 19 这个问题我在physics回答过你了吧？
最常见的思路是把A(x)按照一组正交函数基展开。
假定展开为A(x)=b1P1(x)+b2P2(x)+...
然后对P1(x),P2(x),...积分即可。
因为定义域是无穷到零。可以使用Laguerre函数作为基。
另外一个办法，是做一个 无穷到零 这个区间到 [0 Pi)之间的映射，
然后用Fourier级数展开。后者的优势在于，系数b1,b2,..可以用FFT快速计算。
现在遇到一个矩阵函数的指数积分，从来就没有见过的情况，不知道该怎么办，，
A(x)=| x 2x | 然后对x求exp{A(x)}从负无穷到0的积分，
|2x 3x^2|
以前只知道指数项是Ax的情形（其中A是常数矩阵），现在这个情况，我是不清楚具体
怎么算， 特地请教大家。
另外如果A是正定常数矩阵，是不是可以有∫exp{Ax}dx(负无穷到0)=1/A，就是把A当成
一个数，完全类别通常的指数积分来算？
谢谢大家 |
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w**a
发帖数: 1024 | 20 in physics, we know:
the integration of \exp{i*k*x}dx, i=\sqrt{-1}, on the real line from -\infty
to +\infty
is a delta function I=2*\pi*\delta(k).
The usual way to show this identity is to use
Fourier transform of a delta function then study its inverse transform.
Here is my question: integrate the same integrand, but from x=0 to +\infty.
Shall we get one-half of I (above value)? How to show this? I really want to
use distribution technique to show this. Thank you. |
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s**********n
发帖数: 1485 | 21 So you are asking for a distribution whose Fourier transform is 0 for x<0
and 1 for x>0?
The answer should be (modulo a constant factor)
(i/2)H + (1/2)delta
where H is the Hilbert transform. |
|
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m*****r
发帖数: 3822 | 23 这个就是径向分布函数吧，我也在想这个有没有解析形式
特殊情况可以写出结构因子S(k)，然后fourier transform
c+ |
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c*****p
发帖数: 22 | 24 最后好的当然是 Annals, Acta,Invention,JAMS
其次的要数 Adv, Duke,Publ. Math. IHÉS
不过其他的如 J. Inst. Math. Jussieu, Ann. Fac. Sci. Toulouse Math, J. Inst
. Math. Jussieu , Ann. Inst. Fourier, Compositio Math, MATH ANN ,AMJ, J
REINE ANGEW MATH , COMMENT MATH HELV , MATH Z , ISRAEL J MATH ,NAGOYA MATH J
, MANUSCRIPTA MATH ,PAC J MATH , TOHOKU MATH J ,之间的好坏就分不清了。不过
似乎NAGOYA MATH J , MANUSCRIPTA MATH ,PAC J MATH , TOHOKU MATH J ,这几种比上
面的都要差些。
牛人们出来说说吧。 |
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d********o
发帖数: 59 | 25 The decay rate of the Fourier coefficients depends on how smooth your
function is. Say, your function you want to approximate is n-th continuously
differentiable, then the coefficients of k-th exponential in the expansion
vanishes like 1/(k^n). Thus, if your function is discrete, then it decays
like 1/k; if it's smooth, the coefficients vanish faster than any inverse
power of k, namely, exponentially fast decay. |
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w*q
发帖数: 1544 | 27 谢谢
在compressible n-s中的应用呢？因为还没入门，所以不太懂。望高手指点一下 |
|
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m********8
发帖数: 123 | 29 好厉害，傅立叶可以解非线性问题了，看来藕得放弃数值计算了 |
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a****t
发帖数: 720 | 30 generally, pseudo-random number generating is based on random walks on
certain abelian finite groups. most classical example is following,
computing recursively X_{n+1}=aX_n+b mod(p) for some large prime p. if a=1,
b=+1,-1,0 with prob 1/3 each, with help of discrete fourier transformation,
you need roughly p^2 steps to get randomness.
more detail you can refer persi diaconis's series papers in late 1980's.
also he has a very nice book, group representations in probability and
statistics.
another |
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y**c
发帖数: 133 | 31 Is there anybody can help me to solve this question:
A sphere of radius=R0 has gas in it;
There has a potential V(R)=V0*exp[-(R-R0)/a].
The atoms are colliding in the sphere, and the potential acts on the particles.
The question is:
How to get the Fourier transform of V(t), i.e.
Integral(from -inf to +inf) [V(t)*V(t+tau)]*exp(-i omega t) d tau
you may consider potential is much smaller than the kinematic energy, if otherwise too difficult to solve.
Thank you! |
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B********e
发帖数: 10014 | 32 秘籍第一条：laplace变换（more general fourier transform)是线性的
你的反证法是对di：如果能用于非线性，非线性分析啊，fixed point理论啊都meaning
less了呵呵 |
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g****t
发帖数: 31659 | 33 即使f:R->R的情况，如果点的数目和分布不好的话，
f也是uniform插不出来的。
足够多的等距点的情况最好处理，fourier series就行。
可以参考(苏联出的)数学百科全书第一册。 |
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q********e
发帖数: 1255 | 34 哥，前几个回帖已经告诉你了你的初始边界条件不兼容
后几个回帖告诉你兼容的话就是个trivial的课堂联系，
随便找本under的fourier的书就有practice
这个答案已经很完美了，到你动点脑筋的时候了 |
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B****n
发帖数: 11290 | 35
Usually it is not an easy problem. Even for the famous Fourier series, it
needs a lot of efforts to prove the set of its basis functions is complete.
If you post the form of f(x,t), people here may have better ideas how to
prove completeness. The luckiest thing is if the set {f(x,t),t in R} can be related to some known complete basis.
Yes. If x is in a separable Hilbert space, it can be expressed as an
infinite sum of orthonormal basis functions. If it is orthogonal to any
basis function, then |
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b****e
发帖数: 906 | 36
.
be related to some known complete basis.
非常感谢您的回复。我在尝试着找能成为完备基的基本特征，所以考虑的是普遍的函数
，并没有特定的函数形式。看起来似乎这种完备性应该不难实现，只是难以证明吧，如
果采用有限空间的矢量分解，一般的矢量总是会和矢量基矢有重叠的，那种出现
collinear的情况总还是特例。
从几何上看，似乎一维空间的每一个地方都有那个连续函数族中的某一个加以特别的强
调（例如一堆的高斯，每个高斯出现的最大值的位置都不同），那么很有可能这个连续
函数族就是一维空间的完备基，只是怎么证明和数学表述呢？
同时以上说的情况太片面，譬如Fourier基矢就是在每个地方基本上相同大小，只是疏
密不同。那么，怎么将数学表述包括尽可能多的情况呢？
domain you want to prove completeness. Also, you have to specify the
functional space; for example, it only covers all the continuous functions,
or o |
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s*******n
发帖数: 740 | 38 I think in most american unis, the first year real analysis includes basic
functional analysis(linear operator, hilbert space, L^p space, some fourier
analysis). |
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q********e
发帖数: 1255 | 39 简单点，distritution扩展了函数的定义，通过研究函数作用在test function上的效
果而非研究函数本身使得更多的操作可以meaningfully defined。比如在distribution
意义下的多项式也可以定义fourier变换，比如在研究pde时经典意义下没法直接证明的
东西可以退一步研究distribution意义下的特性，然后考虑更高的regularity等等。
但distribution也要满足一定的性质，比如连续性等，才有意义。
distribution理论里的test function具有紧支集，它们在H_0^1中稠密，所以H_0^1上
的线性泛函都是distribution。而test function 一般而言在H^1_{0,\Omega}里不稠密
，所以后者上的线性泛函不全是distribution。比如一维情况下，假设你的H^1_{0,（0
，1）}包含所有在0点不必等零的函数\psi，那么(u,\psi)=\psi(0)定义一个非零线性
泛函u。 如果u是distribution，u作用在任何test funtion \phi\in D( |
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c*****t
发帖数: 520 | 40 抱歉，我似乎没有把意思表达清楚。
我想问：以MIT 为例，数学系开的名为Wave Propagation的课程，和系里面研究
dispersive equations的教授的工作，似乎差得很远。
是不是要去学习dispersive equations，即便PDE论文的题目是关于具体的方程的，例
如Schrodinger equations ，Water waves 等课题，也是和物理背景没有什么关系，只
是和Fourier transform 等分析技巧有关的？看应用数学方面的专著，例如上述MIT 课
程讲义，以及Mark J. Ablowitz的书，并不能起到什么帮助？
请了解的朋友指教。非常感谢。 |
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s*******n
发帖数: 740 | 41 No need to take courses in undergraduate-level number theory. You can learn
it on your own.
Be sure to take a topology class.
The important courses in my opinion:
Real analysis (basic analysis, measure theory, functional analysis, fourier
analysis)
Complex analysis
Abstract algebra
PDE
ODE
Probability (calculus-based is enough)
Point-set topology and basic algebraic topology
classical differential geometry
differentiable manifolds (advanced differential geometry)
If you are going to do econ, you |
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a*********3
发帖数: 660 | 42 证明：
老子五年前高考就因为数学考了个全班第一！！！！！！
就尼玛进入数学这条不归路了啊！！！！！！！
迎新典礼里院长说数学是最简单的只有最笨的人才会来学数学！！！！
老子真尼玛笨得去学数学了啊！！！！！！
从幼儿园学算术小学中学到大学学了十几年数学还是学不懂啊！！！！！
你们叫嚣着要上一学期数学要学一年数学期末高数考研高数！！！！！
老子大学四年天天都是数学课啊！！！！！考研七门课五门是数学啊！！！
高等代数学完还有近世代数啊！！！！！
解析几何学完还有微分几何啊！！！！！
常微分方程学完还有偏微分方程啊！！！！！！
尼玛复变函数两遍也学不懂实变函数十遍也学不懂啊！！！！有木有啊！！！！
特么泛函分析微分流形老子提都不想提啊！！！！！
尼玛群环域里a+b不一定等于b+a啊！！！！！！
老子学完近世代数连特么四则运算都不敢算了啊！！！！！！
尼玛俩实数加减乘除都重新定义了啊！！！！！
连x和它倒数相乘等于1老子都不会证啊！！！
尼玛说微积分难 你知道你学的是黎曼积分吗！！！！！！！！
你知道还有个勒贝格积分还有个斯蒂尔吉斯积分还有N种各式各样的积分麻！！！！！
尼玛一元的都积不出来给你... 阅读全帖 |
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G********n
发帖数: 615 | 43 Xiang Tang awarded the 2010 Andre Lichnerowicz Prize
September 2010
About the André Lichnerowicz prize in Poisson geometry
The André Lichnerowicz prize was established in 2008 to be awarded for
notable contributions to Poisson geometry. The prize is to be awarded every
two years at the International Conference on Poisson Geometry in Mathematics
and Physics" to researchers who had completed their doctorates at most
eight years before the year of the Conference.
The prize was named in memory of An... 阅读全帖 |
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n******t
发帖数: 189 | 45 class A content(two semester):
Set theory/fundamentals. Axiom of choice, measures, measure spaces, Borel/
Lebesgue measure, integration, fundamental convergence theorems, Riesz
representation.
Radon-Nikodym, Fubini theorems. C(X). Lp spaces (introduction to metric,
Banach, Hilbert spaces). Stone-Weierstrass theorem. Basic Fourier analysis.
Theory of differentiation.
class B content(two semester):
Probability spaces. Distributions/expectations of random variables. Basic
theorems of Lebesque theor... 阅读全帖 |
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s****i
发帖数: 216 | 46 对heaviside function theta(x) 做傅里叶变换得到
-i/w+pi*dirac(w)
那么对theta(x-a)做傅里叶变换理应的到
e^(-ia)*[-i/w+pi*dirac(w)]
但是用mathematica公式
FourierTransform[HeavisideTheta[x - a], x, \[Nu],
FourierParameters -> {1, -1}]
得到的是计算
e^(-ia)*[-i/w]+pi*dirac(w)
这是怎么回事呢？ 以前没学过傅里叶变换， 但是要用到下， 不是很了解 求板上达人
指教 |
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B********e
发帖数: 10014 | 47 what you have is initially
e^(-iwa)*[-i/w+pi*dirac(w)]
and
f(x)dirac(x)=f(0)dirac(x) |
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jl
发帖数: 398 | 49 跟 harr 小波还是不一样的. 我用的 square wave = sign ( sin or cos ).
不知道有没有什么定理 说 跟 sin, cos对应的 waveform 是Complete Orthogonal System? |
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J*****n
发帖数: 4859 | 50
U call it 实在太帅了?
只能说你太狭隘了.....
Also, math only remember two kind of people:
1. Whose theory is applied in research and application, like Newton, Gauss,
Fourier, Laplace etc.
2. Whose problems still are open, like Riemann, Fermat.
No body will remember people who sololy 关闭大门. |
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