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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How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...
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12 views

Calculating h-ellipticity

How do we calculate h-ellipticity $E_{h}$ of standard five point discrete Laplacian of two dimensional partial differential equation?
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92 views

What does the Fourier transform of $1/x^2$ mean?

If I ask Mathematica to compute the Fourier transform of $\frac{1}{x^2}$ using the FourierTransform function, it gives me a result of ...
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1answer
38 views

intuition behind an identity related to fourier transforms

I saw the proof of this identity in a question about Fourier transforms : $F(f(−t))w=F(f(t))(−w)$ Can someone give the intuition behind it ? What I understand of Fourier transform of a function ...
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27 views

Proving something is a convolution operator…

If we define the operator $K(a)=F^{−1}aF$ where $ F:L^2({\mathbb R})\to L^2({\mathbb R})$, is the fourier transform given by $$\left(Ff\right)\left(x\right)=\int_{{\mathbb ...
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1answer
51 views

Fourier transform of exp(cos)

How do I calculate the Fourier transform ($t \rightarrow \omega$) of the following: $\exp(A\cos(\omega_0 t))$ $A$ is a real constant, and $\omega_0$ is a real and positive constant. I know that this ...
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1answer
34 views

a question about Fourier transforms

I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ? $\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$ $if -t=x\to -dt=dx$ ...
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1answer
15 views

2D Fourier Transform proof of Similarity Theorem

I have to solve an exercise, but if i could use the following theorem, it would be piece of cake Similarity Theorem if $ \mathscr{F}\{g(x,y)\}= G( f_x,f_y)$ then $ \mathscr{F}\{g(ax,by)\}= \frac ...
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22 views

Aperiodic signals fourier transform short question?

What is the fourier transform of the aperiodic signals with infinite sequence? How about the transform of aperiodic fourier signals with finite sequence?
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1answer
62 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
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40 views

Is there any commonality between the use of Parseval's Identity in two different contexts?

In Fourier analysis, Parseval's Identity relates to "the summability of the Fourier series as a function." In inner product space analysis, the "identity" works as a "Pythagorean theorem" relating ...
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1answer
11 views

infinite discrete abelian group

I am not too familiar with Fourier analysis, but I needed to use a certain result. I would appreciate any assistance. I was reading a literature in Foruier analysis and it said something like "Every ...
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1answer
35 views

Is there an equivalent of Plancherel's theorem with wavelets transform?

For the Fourier transform, we know that: $$||f||_2=c||\hat{f}||_2$$ where $c$ depends on the normalization. Is there an equivalent with wavelet transform? Thanks.
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1answer
46 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
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2answers
22 views

How to show something is a convolution operator?

I have the operator $W(a)$ defined by $$W(a)=F^{-1}aF$$ where $F$ denotes the fourier transform and $a$ is a function on $L^{\infty}$. I need to prove that this is convolution operator, but I don't ...
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1answer
31 views

When is a function a Fourier transform of an integrable function?

Specifically, in the case $f(\xi)=\frac{1}{(1+\xi ^2)^\epsilon}$ where $0<\epsilon<1$. I wish to prove this is a Fourier transform of a $L_1$ function. Any insight into the manner would be ...
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1answer
48 views

In signal processing, every where you see infinity. Why?

Everywhere, in signal processing you see infinity. For example, in Fouriers, correlations. But no body would live to see infinity. Why do we aritificially talk about infinite time signals and then ...
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1answer
43 views

Proving that $\sum_{n=1}^\infty{[a_ncos(nx)]^{(m)}+[b_nsin(nx)]^{(m)}} \leq \sum_{n=1}^\infty{(|a_nn^m| + |b_nn^m|)}$

Okay so here's the the problem: Let $k \in \mathbb{Z}.$ Then, $\forall$ $m \in [1,k]$, $$f^{(m)}(x) = \sum_{n=1}^\infty{[a_ncos(nx)]^{(m)}+[b_nsin(nx)]^{(m)}} \leq \sum_{n=1}^\infty{(|a_nn^m| + ...
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1answer
56 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
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1answer
47 views

Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)

I have two specific questions. The first is the result I actually need, and the second would let me prove it. EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third ...
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2answers
20 views

How to get the real frequency from an FFT output graph?

I am working in Origin 7.0 and trying to understand how the FFT Analysis works. In an attempt to understand, I decided to run a test of the program using known frequencies. The function I used was: ...
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1answer
20 views

Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
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120 views

Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
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33 views

Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following ...
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convolution of three functions of two variables

Give three functions of two variables $a(x,y),b(x,y),c(x,y)$ one can construct the following convolution like integral: $y(x,y) = \int dx' dy' a(x',y')b(x-x',y') c(x-x',y-y')$ which I have a hard ...
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1answer
42 views

Fourier Transform, and identities

I need to calculate the fourier transform of this $ t \cdot sin(t-3)+2 \cdot t \cdot cos(3t) \cdot rect(6t) $ is the following valid, based on the first fourier identity i've read in a book, and ...
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36 views

Three Dimensional Fourier transform of a raidal function

Scaling analysis shows that the three dimensional Fourier transform of the function $f(\mathbf{r})=1/r$ is proportional to $1/k^2$. On the other hand, when working with spherical coordinates ...
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23 views

Fourier transform theorem

I have an expression: $c(s) = \int_{-\infty}^{\infty} d\omega e^{-i*\omega*s}*F(\omega)$ $G(W)=\int_{0}^{\infty} ds e^{i*W*s}*c(s) $ Is $G(W) = F(W)$? What is the relation of G(W) and F(W)? ...
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25 views

Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
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50 views

Berlekamp Massey and DFT

I was looking into the Berlekamp Massey algortihm, for LFSR, over GF(2) wondering if there was any DFT(alternately FFT), for the above scheme. Also, is there any generalization to Fn, ie, start ...
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1answer
19 views

Fourier transforms similar $\Rightarrow$ functions similar?

I am wondering if there is a theorem that states something like the following? If $$\big|\;\tilde f(\omega)-\tilde g(\omega)\,\big| < \varepsilon\qquad \forall\omega$$ then there exists a ...
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1answer
50 views

Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
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35 views

Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
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19 views

Development of Hilbert transform relationship

In the development of Hilbert transform relationships, Prof. Oppenheim has chosen \begin{equation} ...
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1answer
54 views

Questions about the Fourier series

$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos{(\frac{2 n \pi x}{L})}+b_n \sin{(\frac{2 n \pi x}{L})}) \ \ \ \ \ (*)$$ The symbol $\sim$ has the following meaning: We know that the right ...
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17 views

Rolling $n$ times with an $m$-sided dice. Closed, finite formula for the distribution of the sum? [duplicate]

My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ . It is practically the discrete convolution ...
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1answer
33 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
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1answer
43 views

Fourier transform - epicycles.

I have $f(x) = \sin(x)$. I thought that when I do Fourier Transform and construct epicycles, than those epicycles will draw that $\sin(x)$ function (but this is probably not case with $\sin(x)$, cause ...
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1answer
48 views

Fourier transform of a potential

I need help computing the distributional inverse Fourier transform of the function $1/|x|^2$ in dimension two. The integral makes sense written as \begin{align} 1/2\pi \int_{\mathbb{R}^2} e^{ix\xi} ...
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1answer
86 views

Fourier Transformation: an Animated GIF

Here I found the animated GIF below. I don't get it! Would someone explain it please?
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1answer
40 views

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an ...
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63 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
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Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
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23 views

FFT of simultaneous signal sequences

There is a signal that is received at n antennas in a time period t. So the received signal at each antenna is $$ s_i = x_1, ..., x_t ~~~~~~~ i = 1,...n $$ where x is a feature of the signal. The ...
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35 views

How to solve this heat equation with fourier method

Solve this via Fourier method: $$u_t-u_{xx}=0 \quad\quad 0< x<\pi, \quad t >0, $$ $$u(0,t)=u_x(\pi,t)=0, \quad\quad t \ge 0$$ $$u(x,0)=2\sin\left(\frac{3x}{2}\right) \, ...
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1answer
35 views

What is Fourier Space

I know a some basics stuff regarding Fourier Analysis (Fourier series and Fourier transforms), but I've seen the term "Fourier Space" come up and I'm having trouble finding a definition for what this ...
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37 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
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37 views

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. ...
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1answer
20 views

How to find the Fourier components

I've got a sinusodial signal: \begin{equation} \Delta I=\cos(\Delta \omega t + \varphi)). \end{equation} and I would like to rewrite it as the sum of a cosine and a sine signal(without a phaseterm): ...
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1answer
39 views

Show that the Fourier transform of a a distribution is $C^{\infty}$

I am trying to understand the solution to the following problem: Let $u \in \mathcal{D}'(\mathbb{R}^{n})$ such that $u(x) = c \log(|x|)$ when $|x|>1$, where $c \in \mathbb{C}$. Show that $u \in ...