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 use Fourier transform to solve Fisher-Kolmogorov?

How can I use Fourier Transform to solve Fisher-Kolmogorov Equation in 1D? \begin{equation} u_t(x,t) = u_{xx}(t) + u(1-u) \end{equation} \begin{equation} u(0,x) = \phi(x) \end{equation} with ...
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52 views

An identity involving Gauss sums and convolution

For a Dirichlet character $\chi$ modulo $N$, the Gauss sum attached to $\chi$ is given by $$G_\chi(m) = \sum_{k \in \mathbb{Z}_N} \chi(k) e^{2\pi i mk/N}.$$ Suppose one has an $N$-periodic function ...
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51 views

$\int x J_0(k x)e^{-x^2/2}dx$ Bessel function decomposition of a gaussian

$$\int ^\infty _0 x J_0(k x)e^{-x^2/2}dx$$ The integral above corresponds to fourier transform in radial coordinates. The fourier transform of a 2D gaussian is still a 2D gaussian. So the integral ...
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21 views

Possibly use a Fourier Transform to perform deconvolution

I need to determine the function g(B) in order to determine its prefactors $a_n$. Here's what I have: h(f,B) is a Gaussian function g(B) is the unknown function i(f) is an inhomogeneous function and a ...
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How to extract power of top two frequencies of a spectrum without using an FFT?

What I'm trying to do is see if a particular frequency component becomes dominant (and I don't really know what the dominant frequency is). Therefore, I figured that I can get the top two peaks of ...
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54 views

Discrete Fourier Transform of $\omega^{n(n-1)/2}$

For the sequence $x_0$, $x_1$, $\ldots$,$x_{N-1}$, let $\omega=e^{2\pi i/N}$ and define the discrete Fourier transform as $$X_k = \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x_n\omega^{nk}\,.$$ I'm interested ...
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Operator $T$ commutes with all translations $\Leftrightarrow$ $T$ is multiplication operator

Let $T: L^2(\mathbb R) \to L^2(\mathbb R)$ be an operator that commutes with all translations $f(x) \mapsto f(x-y)$. Why does it follow that the Fourier transform of $Tf$ is then given by ...
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28 views

Mapping properties of a pseudo-differential operator of negative order

Let $H^s$ denote the Sobolev space on $\mathbb{R}^d$. Let $P$ be a pseudo-differential operator of negative order $-m$ where $m > 0$. Let $P^*$ denote its $H^0$-adjoint. Is $P^* P : H^s \to ...
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1answer
45 views

How to Prove Plancherel's Formula?

I have difficulty in proving Plancherel's formula in Fourier transform. Here is what I have thought: In this question, I denote complex conjugation by an overline and (inverse) Fourier transform is ...
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1answer
30 views

Fourier transform with initial limit starting from 0

For a function $f(p)$, the Fourier transform is $\hat{f}(p)=\int_{-\infty} ^{\infty}e^{-i xp}f(x)dx$. What are the conditions that i write it as; $\hat{f}(p)=2\int_{0} ^{\infty}e^{-ixp}f(x)dx$? I ...
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20 views

$\{f\in L^{1} : \hat{f} \in L^{p} \}$ closed under convolution?

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ We note that $L^{1}(\mathbb R) \ast ...
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59 views

Fourier transform of the Heaviside function

As you can see from the title I want to calculate the Fourier transform of the Heaviside function $u(t)$. Proven the the Heaviside function is a tempered distribution I must evaluate: $$ \langle ...
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1answer
60 views

How does one define the Fourier transform of a probability distribution?

Say $p_X$ and $p_Y$ are two probability distributions on a $m$ element set. Then I see an equality written as, $$\sqrt{m} \vert \vert p_X - p_Y \vert \vert _2 = \sqrt{ \sum_{k=0}^{m-1} \vert ...
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1answer
30 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...
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1answer
50 views

Composition operators on fractional-order Sobolev spaces

Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication provided $s > 1/2$. This is proved for ...
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70 views

Is this operator a Fourier multiplier operator?

I want to study the Fourier transform of $$L_{\alpha}(t) = \frac{e^{i\alpha t}}{t^2} - i\frac{\alpha}{t}$$ Basically i am trying to get a grip on, given a $f$, what is $f(t)\ast L_{\alpha}(t)$ and am ...
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1answer
31 views

Question about the limits of definite integrals

Let me take an example that I've come across while studying Fourier series, We all know that $$\int_{-a}^{a} \sin \left( \frac{n\pi x}{a} \right) dx = 2 \int_{0}^{a} \sin \left(\frac{n \pi x}{a} ...
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33 views

Show there exists $n\in \Bbb{Z}$ such that $\lambda =({2\pi n\over T })^4$.

Let $f:\Bbb{R}\to \Bbb{C}$ be an infinitely differentiable function with period T. Also, $f^{(4)}=\lambda f$. Show there exists $n\in \Bbb{Z}$ such that $\lambda =({2\pi n\over T })$. Attempt: I did ...
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Role of i in Fourier transform

I've seen several derivations of the Fourier transform, but most don't cover the conversion to the form $$ S(f) = \int_{\infty}^{-\infty} s(t)e^{-i2\pi ft} \;\mathrm{d}t $$ What is the role of ...
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1answer
34 views

What can we say about the transform of a function on a truncated domain, with respect to the transform on the full domain?

Let $f$ be a function on R and $\hat f$ its Fourier transform. Consider a truncated version of $f$ called $\bar f$ whose value outside an interval is $0$. Formally, $\bar f(x) = f(x) * 1_{x \in I}$ ...
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1answer
36 views

Integration of a generic radial function in polar coordinates

I need to perform the following integral $\int{P(k) e^{i \vec{k}\cdot \vec{\Delta r}} \frac{d^2k}{(2 \pi) ^2}}$ using polar coordinates. I think the result should depend on some Bessel function, but ...
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Question about the Riemann-Lebesgue Lemma proof

Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$ I get all the steps of the proof, except ...
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Confused by certain interpretation of expected value…

I read the following in Stein / Shakarchi's Fourier Analysis book, where they discussed the notion of expectation of a probility density. "Consider the simpler (idealized) situation where we are ...
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32 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz [duplicate]

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
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35 views

Positiveness of partial sums of type $ \psi * D_N $

In his paper about Extremal Functions for the Fourier Transform (see, for example, here? https://projecteuclid.org/download/pdf_1/euclid.bams/1183552525), Jeffrey Vaaler, while trying to build ...
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24 views

About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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58 views

Show there exist a constant $c\in \Bbb{C}$ such that $\int_{0}^{1}|{f-c}|^2<{1\over 36}$

Let $f:\Bbb{R}\to \Bbb{C}$ be a $1$-periodic function, $f\in C^1$ and $\int_{0}^{1}|f'|^2\le 1$. a. Show $\sum_{k\ne 0}|{\hat{f}(n)}|^2\le {1\over 4\pi^2}$ (I did it already, and that question is ...
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The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is non-negative. We can observe that the function $|f|^2$ has a nice property : ...
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1answer
43 views

Fourier transform of periodic function

Is it possible to Fourier transform a periodic function f(x) = f(x+L) with period L, numerically, only over the range x = 0 to L and use periodic boundary conditions to enforce the periodicity of the ...
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52 views

Does the Fourier series converge at $x=0$?

Let $f(x)$, a $2\pi$ periodic funciton such that $f(0) = 1$ and for every $0\ne x\in[-\pi,\pi]$: $f(x) = 1 + \sin \frac{\pi^2}{x}$. Is the Fourier series of $f(x)$ converges at $x=0$? If so, what ...
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1answer
26 views

Euler equation formula

When I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx] $$ I am getting following integrals: $\int_{-\infty}^{\infty} dx f(x) cos(kx)$ ...
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Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
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Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
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Applying the inversion formula to Hardy's theorem

So, in the proof of Hardy's theorem, who says that $\zeta$ has infinite zeros on the critical line, we have eventually that ...
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3answers
53 views

Question about the Fourier Inversion Formula

We have $$\hat{f}(\xi)=\mathcal{F}f(\xi):= \int_{-\infty}^{\infty}f(x)e^{-2\pi i\xi x}dx,$$ with $f\in L^{1}$, and the Fourier inversion formula says that ...
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37 views

Help with Fourier transform of product

I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and ...
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2D Discrete Fourier Transform on an Image - Example with numbers (rgb)

I am trying to write my own function that takes an image, an pixel by pixel it calculates that pixel value that will produce a 2D Fourier Transform image. I have no idea about signal processing, my ...
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Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...
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What is the Fourier transform of Riemann Zeta function?

All: Is there an explicit form of Fourier Transform of Riemann Zeta function ? Also, is there an discrete Fourier Transform (DFT) of Riemann Zeta function ? I remembered I had seen something like ...
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53 views

Fourier transform of exponent

I need to count the Fourier transform of the following function but it does not seem so obvious for me. $f(x)=(e^{-ab})-1$ for $x\ge0$ and $f(x)=0$ for $x<0$ where: $a=1$ and $b=-1$ I don't ...
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$f,g \in R(T)$ such that $\hat{f} \cdot n^{2/3} = \hat{g}$ prove that $f$'s Fourier series converges absolutely.

Can someone help me by checking my solution. Is there a shorter More elegant solution ?(i'm almost sure you can some how express $f$'s Fourier series using $|\hat{g}|^2$ + constant, i saw someone do ...
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165 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
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37 views

Fourier transform of a $H(x)$ product distribution

So I am given this simple example, where $T \in \mathcal{S}(\mathbb{R})$: \begin{equation} T=(\mu +\lambda x+\beta x^2)H(x) \end{equation} where $H(x)\in \mathcal{S}(\mathbb{R})$ (also notated as the ...
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1answer
52 views

Orthonormal Basis of a function

An Orthonormal Family $\{e_k\}_{k\in\mathbb{N}}$ is a basis if and only if $$f=\sum^\infty_{n=1}\hat{f}(n)e_n \ \ \ \text{in} \ \mathcal{L}^2(\mathbb{R})$$ where $f\in\mathcal{L}^2(\mathbb{R})$ ...
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1answer
31 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
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A convolution equation with two unknowns

I consider the following convolution-type equation with two unknowns $f_1$, $f_2$: $$ a_1 * f_1 + a_2 * f_2 = 0, $$ where $a_1$, $a_2 \in L^1(\mathbb R)$ and $*$ is the ordinary convolution. This ...
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68 views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
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48 views

How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
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1answer
32 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...
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25 views

Fractional derivative of $e^{-x^2/2}$ using Fourier transform and Taylor series

I am not familiar with fractional calculus, so I want to know what I am doing wrong. The convention I use $$\int^\infty_{-\infty}e^{-\frac{x^2}{2}}e^{-i k x}dx=\sqrt{2 \pi}e^{-\frac{k^2}{2}}$$ I am ...