Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).
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Confusion with Fourier series - please check my notes.
I've been having some trouble with calculations involving Fourier series, despite understanding the concept itself because of the multitude of cases: symmetric/asymmetric intervals with different ...
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2answers
31 views
Inverse fourier transform of exponentially decaying function in the frequency domain
I want to take the inverse Fourier transform of the following function:
$$ \hat{f}(\omega) = \begin{cases}e^{-r \sqrt{\omega}} & \text{for } \omega > 0 \\ 0 & ...
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1answer
38 views
Fourier transform of text
How could I/is it possible to take a fourier transform of text? i.e. What domain would/does text exist in? Any help would be great.
NOTE: I do not mean text as an image. I understand it's value, but ...
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1answer
29 views
convolution of Poisson kernel with itself
I am trying to prove that $p(l,x)*p(u,x)=p(l+u,x)$ where * denotes the convolution of two functions and $p(l,x)=\dfrac{l}{\pi(l^2+x^2)}$. I am having trouble in integrating the left hand side.
...
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0answers
16 views
Convert to displacement time signal (from accelerometer) using velocity time signal or displacement frequency signal
Firstly, I try to use ifft to convert displacement frequency into displacement time but the ifft fails to give the original sign.The code as below:
...
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24 views
What should i be getting from the DFT?
I was attempting to do a discrete Fourier transform through a computer program on a list of numbers. Before doing that I decided to test it by running through a list of 1000 numbers which I created by ...
4
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57 views
Is this a spectral decomposition/embedding/isometry?
Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$.
Now if I take the same ...
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1answer
30 views
Show that the subspace A is the whole Hilbert space H
"Let $A$ be a subset in a Hilbert space $H$, such that $x\in H$ and $x \perp A$ imply $x = 0$.
(1) Show that the closed subspace that is generated by $A$ is $H$.
(2) Let $f(x)$ be a square summable ...
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1answer
35 views
Fourier Properties
If I have a function $h(x)$ defined as $f(x)\ast g(x)-n(x)$, can I just say that $H(\omega)=F(\omega)G(\omega)-N(\omega)$ Where the capital letters are the fourier transforms. I am familiar with the ...
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1answer
22 views
Fourier transform of a 6th order ODE
How can I take the Fourier transform of a $6^\text{th}$ order ODE, which includes terms such as $\tfrac{1}{r}D^6 U(r)$ or $\tfrac{1}{r}D^3 U(r)$? Here $r$ is the independent variable and $U(r)$ is to ...
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24 views
2D Spatial - Frequency Domain Results Not matching?
I've got a process I'm performing in the spatial domain as:
1. Create a grid of points, 2. Take a square of some Length and Width and place it on the x-y axis origin 3. Count the number of points ...
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40 views
Dirchlet Problem and fourier series
I apologies for not using latex. I have 3 images attached to this question. The first two are the background and the last image has the question. The question is in the third image under problem 3.
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2answers
47 views
Are the Fourier sine and cosine transforms commutative?
That is to say, is it true or false that
$$\mathcal{F}_c(\mathcal{F}_s(f(x)))(\xi)\equiv\mathcal{F}_s(\mathcal{F}_c(f(x)))(\xi),$$
and if they are not then are there any conditions on $f$ for which ...
0
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0answers
34 views
Fourier transform
I am new to the distribution theory and have some difficulties to calculate curtain fourier transforms. Can you help me with $\frac{e^{-xb}}{x+i0}$. I got to the point $\lim_{\epsilon \to ...
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1answer
38 views
Inverse Fourier Transform of $\prod_{j=1}^n \frac{k_j}{k_j+i\omega}$
I'm trying to work out the inverse Fourier transform of
$$F(\omega)=\prod_{j=1}^n \frac{k_j}{k_j+i\omega}$$
with $k_j \in \mathbb{R}^+$ and using the definition of the Fourier transform where
...
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0answers
41 views
Fourier transform of a tempered distribution
anybody knows how to calculate the fourier transform of $e^{-ax}, a>0$ in the sense of tempered function. I manage to find out it is $ \delta $ (y+ia) but it does not seem right as the 'argument' ...
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1answer
23 views
questions about proof of reverse formula of Fourier transform
We had the following theorem in class for a fourier transform $\widehat f$:
Let $\widehat f$ be the restriction of a $\mathbb C$ definied meromorphic function $F$. Let $F$ have a finite number of ...
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1answer
34 views
Edge effects in 2D fourier transform?
I am taking the following image
then applying Fourier transform to it (discrete, FFT), then removing high frequencies from it and finally converting it back to image.
The result is following
...
2
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1answer
36 views
Plancherel theorem of fourier series.
I want to show Plancherel's theorem: If $(c_n)\in l^2(\Bbb Z)$, there exists a unique function $f \in L^2[-\pi,\pi]$ such that $\hat{f}(n)=c_n$.
I saw a proof goes like this: $f_k=\sum_{|n| \le ...
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0answers
14 views
How can you see which points in the spectrum is from which pixel in the original image? [migrated]
Take the image and spectrum below.
If I look at the spectrum, it just look like noise....
How to make sense of it intuitively?
Image:
Frequency spectrum of image (using Fourier Transform):
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1answer
48 views
Convergence of fourier series in $L^2$
I want to prove Carleson's theorem for $L^2$: If $f\in L^2(\Bbb T)$, $$\lim_{n \to \infty}S_n(f)=f \ \ \ a.e.$$
I have learned that $$\lim_{n\to \infty}||f-S_n(f)||_2=0$$
Take limit into the norm, ...
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1answer
27 views
Inverse Fourier transform of characteristic function
Let $Q$ be a measurable set in $\mathbb{R}^2$
Let
\begin{equation}
1_Q(\textbf{x}) = \left\{ \begin{array}{ll}
1 & \mbox{if $\textbf{x} \in Q$},\\
0 & \mbox{otherwise},\end{array} \right.
...
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2answers
36 views
How to remove high frequncies having fft?
What are the units of measure of FFT elements?
Can I just set higher elements to zero to filter out higher frequencies? It looks like no.
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3answers
64 views
when Fourier transform function in $\mathbb C$?
The Fourier transform of a function $f\in\mathscr L^1(\mathbb R)$ is
$$\widehat f\colon\mathbb R\rightarrow\mathbb C, x\mapsto\int_{-\infty}^\infty f(t)\exp(-ixt)\,\textrm{d}t$$
When is this indeed ...
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4answers
134 views
How to interpret Fourier Transform result?
Can anybody tell me what result of discrete fourier transform means? I know all theoretical stuff and pretty graphs, that it is a change of domain from time to frequency and so on.
But I want to ...
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1answer
31 views
derivative of parameter integral in $\mathbb C$
Let $f:\mathbb R\rightarrow\mathbb R$ be continuous and let $g(x):=xf(x)$ be absolutely integrable. Then $\widehat f'=-i\widehat g$.
I know this would be true if I can differentiate in the integral ...
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1answer
30 views
How do you invert a characteristic function, when integral does not converge?
I need to find the probability density of some distribution with characteristic function given by:
$$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$
I know the formula for inverting a ...
0
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1answer
32 views
Fourier transform of periodic signal
I have a question that is similar to this one but slightly different.
If I have discrete signal $$s(t) = \sum_k n_k \delta(t-kT_0),\quad k=0,1,\dotsc,$$ where $n_k$ are just some scalar numbers. What ...
4
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1answer
48 views
derivative of fourier transform
Let $f\in C^k$ and $f^{(k)}$ be absolutely integrable. I want to show for the fourier transform:
$$\widehat{f^{(k)}}(z)=(iz)^k\widehat{f}(z)$$
I want to prove it for $k=1$ and did the following:
...
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1answer
51 views
Convolution of distributions.
We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
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1answer
47 views
Fourier series representing a continuous function?
I am fairly sure the answer to my question is "No", so this is more of an affirmation/reference request question.
Given a Fourier series $\sum\limits_{k \in \mathbb{Z}} a_k e^{kxi}$, we can view it ...
3
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2answers
56 views
Problem of convolution.
If we are given with a polynomial $\mathcal P$ and a compactly
supported distribution $g$. Can we prove that their convolution will
be a polynomial again?
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1answer
58 views
Fourier transform of a function of $\cos(\omega \cdot t)$
I need your help.
Suppose I have a function $V(x)$.
Now suppose I vary it with time as $W(x,t) = V(x+\cos(\omega \cdot t)$.
I need to find out What is the fourier transform of $W(x,t)$ with respect ...
2
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2answers
90 views
Solve Laplace equation in the upper half plane
I need to solve
\begin{eqnarray}
u_{xx} + u_{yy} = 0 \quad \quad y>0 \quad -\infty < x< \infty
\end{eqnarray}
With boundary condition
\begin{eqnarray}
\frac{\partial u(x,0)}{\partial y} = ...
2
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1answer
65 views
Paley-Wiener Theorem
In Proposition 5.3.11 of Bratteli-Robinson Vol II the Paley-Wiener Theorem is stated as follows:
Is this the correct statement? Aren't $f$ and $\hat{f}$ mixed up? I know the following version of ...
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1answer
40 views
Confusion related to DFT [closed]
I have this confusion related to how to reconstruct a signal after performing DFT on it.
Lets say I have a signal have values
x=1:100
Now if I apply the FFT on this signal x=1:100 and I remove 50 ...
0
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1answer
41 views
Do you need to zeropad an image of 1920*1080 to 2048*2048 when using the Cooley-Tukey FFT?
User @Paul_R wrote that you need to zeropad an image of 1920*1080 = 2^20,984
to 2048*2048 = 2^22 when using the Cooley-Tukey FFT?
Why don't we just zeropad it to 2^21=2048*1024?
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1answer
24 views
Does the 'FFTW' have a cost of $k*N*\log_2(N)$?
@Michael C. Grant wrote that "the cost of FFTW isn't an easy formula based on log_2 anymore."
But the Wikipedia article says that the FFTW (Fastest Fourier Transform in the West) "can compute ...
1
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1answer
30 views
how can I get the continuity?
I'm reading 'Foruier analysis methods for PDE's'by R.Dancin.On page 43, at the end of thereom 2.2.3,to prove u belongs to $$C([0,T];\dot{B}_{p,r}^s)$$,the author used the density of ...
3
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0answers
32 views
transform that is invariant under rotation
We know that the magnitude of the Fourier transform (resp. Mellin transform) of a shifted (resp. scaled) function is identical to the magnitude of the original function. I wonder if there is a ...
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22 views
using Paley-Wiener to get support and then estimate inf sup
I try to make a complete question out of my previous ones.
Define the function
$$ \tilde{f}_n(\omega)=\frac1{\sqrt{2\pi}} \frac{\sin R\omega/2}{R\omega/2} s_n(R\omega/2\pi),$$
where (using Weierstrass ...
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1answer
34 views
Fourier Transform from Discrete Fourier Transform
If I have the basic Discrete Fourier Transform from a discrete function $x[n]$, like this:
$$\displaystyle X[k] = \sum\limits_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn} $$
How can I get to the expression ...
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1answer
57 views
How to show $C_p^k([-\ell, \ell])$ is not a Banach space?
I need to show the space $$C_p^k([-\ell, \ell])=\{f\in C^k(\mathbb R; \mathbb C); f(x+2\ell)=f(x), \forall x\in\mathbb R\},$$ is not a banach space with the norms ...
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0answers
37 views
Fourier transform of convolution in a finite range
Can anyone help me evaluate the Fourier transform of of the following function,
$t \in \mathbb{R}$, $\lambda \in \mathbb{C}$, $g:\mathbb{R} \rightarrow \mathbb{R}$,
$f(t) = \int_{t_0}^t ...
1
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1answer
47 views
Calculating Fourier Transform of $1/|t|^n$
I have found the Fourier Transform of $x(t)=|t|^{n}$ and i can't calculate the Fourier Transform of $x(t)=|t|^{-n}$. Any suggestions?
3
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1answer
81 views
Fourier transform of $\text{sinc}^3 {\pi t}$
$$f(t)=\frac{\sin^3(\pi t)}{(\pi t)^3}$$
I want to calculate the Fourier transform.
I can't calculate this integral:
$$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t$$
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0answers
12 views
Separating waves of very close wavelengths [migrated]
I'm new to wave problems, so excuse my simplified words.
I have a signal consists of high range of frequencies . I applied a Band-pass filter to this signal and I have been able to separate it to ...
0
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0answers
41 views
Fourier, the Fourier transform
Could You help me? Where
$g(t)$ is Cantor function:
$$G(\omega)= \int_0^1 e^{2\pi i\omega t}dg(t)$$
Show, that $G(\omega)\not\to0$, if $\omega\to\infty$
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1answer
24 views
Convergence of a series of a given metric..
I'm trouble with a metric defined over a given set: Consider $\mathcal{P}=C_p^\infty([-\pi, \pi])$, that is, $\mathcal{P}$ is the set of all infinitely differentiable functions $f:\mathbb R\rightarrow ...
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39 views
Fourier transform for even function
How find fourier transform for $$f(x) = {\mathcal{X}}_{[- \frac{1}{2}, \frac{1}{2}]} \cdot \cos^n{ \pi x}, n \in \mathbb{N} , x \in \mathbb{R}
$$ where $$
{\mathcal{X}}_{[- \frac{1}{2}, ...
