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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Distribution with finite support.

If $f \in\mathcal D′(\Bbb R^n)$, is a distribution and support of this distribution is a set of finite points in $\Bbb R^n$. Can anyone tell me that what will be the general form of this ...
3
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1answer
89 views

Differential equation on $\Bbb R$

We have a differential equation on $\Bbb R$ of the form $$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$ where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a ...
3
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3answers
142 views

$\hat{f}(x)=\frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty}{f(x)e^{-i\omega x}dx}$ and $\hat{f}(x)=\int_{-\infty}^{\infty}{f(x)e^{-2\pi ixy}}dx$

Fourier transform - difference between $$\hat{f}(x)=\frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty}{f(x)e^{-i\omega x}dx}$$ and $$\hat{f}(x)=\int_{-\infty}^{\infty}{f(x)e^{-2\pi ix\omega}}dx$$ Can you ...
2
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1answer
95 views

This is an expression in my Fourier Analysis text book, but I cann't understand it.

This is an expression in my Fourier Analysis text book, but I cann't understand it. I have done something,but I can't go on. Is there something wrong with the expression in my text book Or my work ...
5
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3answers
194 views

Let $f:[a,b]\to\mathbb R$. Evaluate $\lim_{n\to\infty}\int_a^bf(x)\sin(nx)\,dx$ [duplicate]

Let $f:[a,b]\to\mathbb R$. Evaluate $\lim_{n\to\infty}\int_a^bf(x)\sin(nx)\,dx$. $f$ is continuously differentiable. I'm told this can be done using basic calculus. It's difficult for me to see ...
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2answers
365 views

Fourier Integral problem

Let $f(x)= \begin{cases} 3, & x \in [0,\pi]\\ 2x, & x \in(\pi,2\pi] \\ 0, & x > 2 \pi \end{cases} $ Express $f$ as a Fourier integral. I don't know which type of integral I'm supposed ...
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3answers
102 views

Fourier analysis on finite abelian groups

Can someone help me show if $f$ is a character of a finite abelian group then for all $a\in G$, $$\sum_{[f]}f(a)\stackrel{}{=} \begin{cases} |G| & \text{if $ a$ is the identity} \\ 0 & ...
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0answers
362 views

Help with Hermitian Symmetry and its inverse Fourier transform in MATLAB.

I have tried to impose Hermitian symmetry on the complex number $z$ which is varies with $x$. I need to take its inverse Fourier transform. A hermitian symmetry should give a real valued inverse FT. ...
10
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1answer
2k views

Taylor Series and Fourier Series

Taylor series expansion of function, $f$, is a vector in the vector space with basis: $\{(x-a)^0, (x-a)^1, (x-a)^3, \ldots, (x-a)^n, \ldots\}$. This vector space has a countably infinite dimension. ...
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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66 views

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
511 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 & ...
2
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1answer
277 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
205 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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212 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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112 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
65 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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2answers
89 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 ...
0
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1answer
25 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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1answer
147 views

Dirichlet 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. ...
5
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2answers
119 views

Are compositions of 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 ...
3
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246 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 ...
4
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1answer
123 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
118 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' ...
2
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1answer
86 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 ...
2
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1answer
461 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
241 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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1answer
70 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
244 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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232 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. ...
4
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3answers
80 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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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 ...
0
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1answer
49 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 ...
2
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2answers
271 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
55 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
150 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
377 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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0answers
108 views

Does the Fourier matrix $F_n$ represent a (tensor) multiplicative function?

At "Complex Hadamard Matrices", I found that, two Kronecker (tensor) products of Fourier matrices $k_1$ and $k_2$ are equivalent, if and only if $k_2$ can be obtained from $k_1$ by a combination of an ...
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1answer
62 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 ...
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2answers
114 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
285 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
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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
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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
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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?
0
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1answer
37 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 ...
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1answer
34 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
73 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 ...
2
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49 views

using Paley-Wiener to get support and then estimate inf sup

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 the Weierstrass product representation for $\sin$) $$ s_n(w) = ...
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1answer
114 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 ...
2
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1answer
122 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 ...