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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Proof by construction of Wiener's tauberian theorem on $\mathbb{R}^n$

Would anyone have a reference to a proof by construction of Wiener's tauberian theorem in $\mathbb{R}^n$? As a reminder the theorem goes as follows: Theorem (Rudin, Functional Analysis Th. 9.5): ...
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Fourier transformation of the symmetric group $S_3$

I am trying to compute the Fourier transformation of the symmetric group $S_3$ following the section 4 of Quantum Computing and the Hunt for Hidden Symmetry. The multiplication table of $S_3$ is as ...
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35 views

an application of the Poisson summation formula

It is written in a paper that I was reading that "by an application of the Poisson summation formula" we have $\sum_{n \ne 0} |n|^{-1} e^{inx} = C \ln |x| + \phi(x)$ with some smooth function ...
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Fourier transform of $e^{-\delta (x)} \cos\left(\frac{1}{x}\right)$

I have a function $$ f(\omega) = \exp\left(-\frac{\gamma}{\gamma^2+\omega^2}\right)\cos\left(\frac{\omega}{\gamma^2+\omega^2}\right), $$ and I'm trying to calculate its Fourier transform at the ...
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22 views

Calculate Norm Operator

I'm trying to solve this exercice: Let $\omega(y)=y^{-4}$ and $L^{1}(\mathbb{R},\omega)$ the space of measurable functions $g:\mathbb{R}\rightarrow\mathbb{R}$ so that $g\omega$ is Lebesgue ...
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Does inverse of all Fouriers transforms have a corresponding function in time domain?

I am trying to cancel out the following transfer function of a system: $$\frac{( 1 - e^{(i*k*T)} ) }{ (i*k)}$$ I thought it would work if I find the inverse Fourier transform of $$\frac{ (i*k)}{( 1 ...
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23 views

Kernel of Helmholtz Equation on a plane

On the $z=0$ plane I have the boundary conditions $V=\delta(x)\delta(y)$ I want to solve for $z>0$. Helmholtz equation is $\nabla ^2 V +k^2 V=0$ I though that spherical harmonics would be useful. ...
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36 views

What is the Fourier transform of $1/|x|$?

I looked it up in several tables and calculated it in Mathematica and Matlab. Some tables say that the answer is simply $$\frac{1}{|\omega|}$$ and in other table it is ...
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47 views

Fourier transform of $1/x$, $x>0$

I'm new to Fourier transform. From Does the Fourier Transform exist for f(t) = 1/t? ,I saw that the Fourier transform of $1/x$ is $\text{sgn}(x)$. Define $$f(x)=\begin{cases} \frac{1}{x} ...
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37 views

Why does inputting complex exponentials into a system give its frequency response?

Let's say I have an FIR filter with the equation: $$ y[n] = \sum_{i=0}^{N-1} h[i] x[n-i] $$ I know that to find the frequency response of this filter, I need to input a complex exponential in place ...
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34 views

Can someone explain negative frequencies when doing the Fourier transform?

I apologize if this question has been asked before. I have looked and have not found a clear explanation. When doing the discrete Fourier transform (e.g. fft in MATLAB) for a vector of discrete time ...
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31 views

Fourier sums convergence

I am given the Sin series expansion: (1)$$x=2\sum_{1}^{\infty}\frac{(-1)^{n+1}}{n}Sin(nx)$$ But how do I deal with a solution given by: ...
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27 views

Littlewood-Paley theorem at endpoints

Littlewood-Paley theorem says that the $L^p$ norm of the square function associated with $f$ is equivalent to the $L^p$ of $f$ when $p\in (1,\infty)$. I'm interested in the endpoint case. Why it ...
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22 views

prove that if $f(x)$ is real, then $\left | F(s) \right |^2$ is an even function

I understand that the fourier transform of f(x) can be broken up into odd and even parts such that the transform of $f(x)$ can be represented by $$F(s) = 2\int_{0}^{\infty}E(x)cos(2\pi x s)dx - ...
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44 views

Numerical integration with FFT

Suppose I am faced with the following integral: \begin{equation} F(\bf{x}) = \int \frac{d^{2}\bf{k}}{(2\pi)^{2}} exp(-i\bf{k}\cdot \bf{x})f(\bf{k}) \end{equation} where $f$ is some (known) function ...
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33 views

Computing Fourier transform of $e^{-ax^2 - b|x|}$

recently I've been working on some estimation problems in $L_2$ space and came across the problem of computing the following Fourier transform for constants $a,b>0$: $$ F(w) := ...
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49 views

Expanding Fourier Series of $f(x)=x^2$ where $0<x<1$ (even and odd)

I tried to solve Fourier series (which appeared on title) and ended up to below solution : on even state : $ \phi(x)= \begin{cases} x^2 & 0<x<1 \\ x^2 & -1<x<0 \end{cases} $ ...
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18 views

What is $ \mathcal S ( \mathbb R_+ )$?

Does Schartz space of functions defined on positive halfline exists in a meaningful way? How does such functions look like and do you know any applications of such space?
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43 views

Fourier transform step-by-step example: $f(x) = 1/2$ where $x\in[0,1]$ otherwise $f(x) = 0$

I'm trying to understand the general procedure for finding the fourier transform of a function f(x). I've seen the general theory, but feel It would help with a concrete example to see how it is ...
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18 views

Fourier Transform of Distribution Equal to the Distribution Itself

We define $T(t)=a\delta^{(n)}+bt^n$ for $a, b$ nonzero complex constants and $n$ a nonnegative integer. I want to find the combinations of $a, b, n$ such that the fourier transform of $T$ is equal to ...
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35 views

CT fourier Transform Proof

I am confused trying to understand the Proof of Fourier Transform from Oppenheim book Signals and Systems. I am pasting the equations directly from the book: ...
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25 views

Criterium forSubspace of tempered distributions

I have a question concerning the subspace $\mathcal{S}'_h$ of tempered distributions defined by $u\in\mathcal{S}'_h\Leftrightarrow\lim_{\lambda\rightarrow\infty}\Vert\theta(\lambda ...
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36 views

Convergence of the sum of fourier coefficients

I have a function $f$ defined on $[-\pi, \pi]$, $f(\pi) = f(-\pi)$, and has continuous first derivative. How can I prove that the sum of the absolute value of the Fourier coefficients converges? The ...
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38 views

Finding a Fourier pair satisying some conditions

I want to find a Fourier pair $(r,\hat{r})$ satisfying some conditions listed below and making $\hat{r}(0)$ as small as possible. The requirements for $(r,\hat{r})$: $r,\hat{r}\in L^1(\mathbb{R})$, ...
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35 views

Eigenfunctions of a 4th Order PDE on 2D domain

What technique would you use to find the eigenfunctions of this problem? $$\nabla^2(\nabla^2 u) = 10$$ $0 < y < G$ $0 < x < L$ With Pure Homogeneous Dirichlet boundary conditions and ...
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29 views

Piecewise linear approximation of a (low order) trigonometric polynomial, with quantization

The Problem Given a trigonometric polynomial of order $K$: $$y(t)=\sum_{k=-K}^K c_k \ e ^ {j k \bar \omega t} \ , \qquad c_{-k}=\bar c_k$$ we want to find the best approximation to it using a ...
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123 views

wave front set - directions of singularities

I am learning about the wave front set of a distribution but am having difficulty understanding some details, which to me seem counter intuitive. We know the fourier transform of a smooth function ...
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46 views

Fourier transforms of distributions

I am reading a proof claiming that every partial differential operator $P(D)$ has a fundamental solution $E$. It says that "if we have a distribution $u$ on $R^n$ with $u(P(D)\phi)=\phi(0)$ for ...
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71 views

A continuous function whose Fourier Series diverges

I have read several times that there exist many continuous functions whose Fourier series diverge at some points (sometimes even on a dense subset of the domain!). I tried to find an explicit example ...
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I have derived a short proof that answers 'DTFT assumes the sequence to be periodic and hence the expression for the same'. Does my proof make sense?

Define: Function '$f$' defined at all (-oo, oo). $U(to)$ is a step function that equals 1 when $t \geq to$ $\sum_{n \epsilon Z}\delta(t - nT)$ is an impulse train that is non-zero for all $t = nT|{n ...
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35 views

Question about Fourier Transform?

Could someone please explain how they got from the first step to the next? I have no idea how the second step follows...
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61 views

The derivative of the trigonometric polynomial

For $n\in\mathbb{N}$, let $\varphi:\mathbb{R}\to\mathbb{R}$ be a $4n-$ periodic function s.t. $$\varphi(t)=n-|n-t|,\quad -n\leq t\leq 3n$$ For each $j\in\mathbb{Z}$ define $$c_j=\int_0^1\varphi(4n ...
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Derivation of First order derivation of Fourier transform

Let $R$ be $(-\infty,+\infty)$. Starting with: $$x(t)=\int_{R}X(f)e^{2i\pi ft}df$$We have its Fourier transform: $$F\{x(t)\}(f)=X(f)=\int_{R}x(t)e^{-2i\pi ft}dt$$ Its derivation is derived as ...
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Fourier Transform of a CFT two point function

In the study of Conformal Field Theory in physics, one encounters the following function $$ \left( \frac{1}{\sinh(x+i\epsilon)} \right)^{2\Delta}. $$ It appears as the correlation function of certain ...
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Find the Fourier Sine series of $x-\frac{1}{2}$ on the interval $0<x<1$

Two Parts to this question are: (a) Find the Fourier Sine series of $x-\frac{1}{2}$ on the interval $0<x<1$ (b) Let $a \notin Z$ and find the Fourier Cosine Series of $\cos ax$ on the ...
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Fourier transform on Laplace equations

We can solve some Laplace equations on the lectangular area by using Fourier transform. However many textbooks only introduce the cases when the area is given as a half plane or a strip. ; $y>0, ...
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Generalization of a FFT with powers of 3

I'm not satisfied with the current answers asked about this on MathSE. So I'm going to ask: Describe the generalization of the FFT algorithm to the case in which n is a power of 3. What's the ...
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problem 9.26 from Folland's real analysis Fourier Transform of $ G(x,t) = (4t\pi)^{(-n/2)} e^{{-|x|^2}/{4t}} \chi_{(0,\infty)}(t)$

I was just given this question from Folland's real analysis second edition dealing with tempered distributions and their Fourier transforms Exercise 26 on page 300 : On $ R^n \times R $ let $ ...
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Extension of the Fourier transform, proof-read

I'm writing a Bachelor-thesis in mathematics which is to be submitted in a couple of days, and would be thankfull if the following arguments concerning the extension of the Fourier transform from the ...
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64 views

Techniques in analytic number theory

I'm fairly new to the subject and trying to figure things out. Would be nice to hear some ideas and trickery for what follows. Suppose we wish to show there exists an integer $x$ in some finite set ...
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Autocorrelation of a signal known only on an interval of finite size

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
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36 views

Zero padding property of FFT

I wonder if the following basic property I thought up is a real property of FFT (or more specifically the discrete version of Fourier series), and if so what is it officially called? The property ...
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Inverse Fourier Transform involving inverse square root

I'm currently working on this paper: http://web.calstatela.edu/faculty/rcooper2/article.pdf and I want to proof Lemma 3.0 in the case of $n=2$, on page 441. It seems that the Ph.D. thesis the author ...
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Fourier transform of $1 - \cos(xe^{-x^2})$

Is there a closed form expression or maybe an infinite series? If not is there a "good" approximation to it? Even a "good" approximation of the fourier transform close to zero frequency would do. Can ...
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Are the Spherical harmonics the S^2 equivalent of the exp(i \pi n) function series?

As I understand it, the Spherical harmonics and the "Fourier functions" $\exp(i\pi n)$ with $n\in\mathbb{N}$ have much in common: Both are eigenfunctions of the angle part of the Laplace operator. ...
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139 views

Fourier series of half of $\sin(\pi x)$

So my question is: Find the Fourier series (using integrals) for the half wave rectified sine function: $$f(x)= \begin{cases}0&-1<x<0\\ \sin(\pi x)& 0<x<1\end{cases}$$ ...
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110 views

A discussion on fourier and laplace transforms and differential equations …?

i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of ...
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Fourier transform of the Cosine function with Phase Shift?

How can i calculate the Fourier transform of a delayed cosine? I haven't found anywhere how to do that. This is my attempt in hoping for a way to find it without using the definition: $$ x(t) = ...
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What are the statistics of the discrete Fourier transform of a Bernoilli process?

The problem I would like to understand the statistics of the discrete Fourier transform of a sequence of uncorrelated events $\{x_n\}$ each of which takes the value $\pm1$ with probability $1/2$. In ...
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54 views

Diagnalization of block matrix with circulat blocks

I have the following Matrix $A = \begin{pmatrix} X \\ Y \end{pmatrix}$ Where X, and Y are circulant Matrices. I want to diaganlize $AA^T$. I tried the following: $AA^T = \begin{pmatrix} ...