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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Fourier integral representation. [on hold]

What is the Fourier integral representation of $$ \begin{cases} f(x)=0, & x<0 \\ \frac{1}{2}, & x=0 \\e^{-x}, & x>0 \end{cases}.$$
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2answers
29 views

Fourier series, instantly determining $b_n$ once $a_n$ is found.

Find the Fourier series of the following function: $f(x) = \left\{\begin{align} 1+x,\quad -1\lt x \lt 0 \\ 1-x,\;\;\;\quad 0\lt x \lt 1\end{align} \right.$ $f(x+2) = f(x),\quad\quad -\infty \lt x ...
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15 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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1answer
33 views

How did Fourier series lead to the development of rigorous analysis?

Once I've heard that the studies of Fourier series have lead to rigorous definitions of such concepts as function, convergence, integral, limit. And also that Cantor's study of Fourier series led him ...
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17 views

Fourier transform and conjugate variables

When you make a Fouriertransform of a function of time $f(t)$, it is said that it's Fouriertransform is a function of frequency $\widetilde{f}(\omega)$. The same argument goes for position and ...
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23 views

Fourier transform with the function in denominator

I'm trying to do a Fourier transform of this function $$\frac{\bigtriangleup f(\textbf{x})}{1+f(\textbf{x})}$$ in terms of $\mathcal{F}(f(\textbf{x}))$. (Just like here ...
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1answer
25 views

Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
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1answer
22 views

$\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$

I can't compute this $\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$. I have separate it into 2 integrals but i can't continue.
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1answer
29 views

Find inverse Laplace transform of $H(s)=\frac8{s^4+4}$

How can we find the inverse Laplace transform of the function $$H(s)=\frac8{s^4+4}?$$
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17 views

How to determine if someone understands Fourier transforms

I barely understand Fourier transforms. I'm looking for a simple test or Q&A that when used would tell me if someone else understood Fourier transforms. How would I know if the person genuinely ...
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23 views

White noise, how is its definition sensical

White noise is defined as as noise containing all frequencies. Now, consider the inverse fourier transform of white noise, $R$ being the fourier transoform of the noise: $$\int_{-\infty}^\infty R ...
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1answer
30 views

Fourier Transform solution of $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - u$

Let $u(x,t)$ solve the partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - u$$ where $x,t\in\mathbb{R}$ with $t>0$ and initial condition $u(x,0) = ...
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16 views

Polynomial division/deflation with FFT

There is a need to divide a polynomial $p(x)$ by polynomial $q(x)$, whereas it is known that the remainder will be zero (i.e. the question is about polynomial deflation). A known method is to use the ...
2
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1answer
31 views

Help solving $\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$ using Fourier transforms

I am trying to solve $$\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$$ $x\in\mathbb{R},\:t >0$. Subject to the conditions $u(x,0)=f(x),\:u,\:\frac{\partial u}{\partial ...
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1answer
33 views

Fourier transform and residue theorem

I want to calculate the fourier transform of the funcion: $$f(x)=\frac{1}{a^2+x^2}$$ So I did: $$\hat f(w)=\frac 1 {\sqrt{2\pi}} \int_{-\infty}^{\infty} \frac {e^{-iwx}}{a^2+x^2}dx$$ And I tried to ...
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1answer
19 views

An integral converges- Analysis Fourier

I want to show that if $k\geq2$ and $f\in C_{c}(\mathbb{R})\cap C^k(\mathbb{R})$ then $\int\limits_{\mathbb{R}} |\xi|^{κ-2}|\hat{f}(\xi)|dm(\xi)<\infty$. I can't understand what can i use for ...
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51 views

Mellin transform on $\mathbb{Z}[\omega]$

Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique factorization domain) are of the forms $a+b\omega$ where $a$ and $b$ are ...
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6 views

Find the spectral density of white noise $\omega$~(0,2)

I am find the spectral density of a white noise given by $\omega$ ~ (0,2). Could you help me to find it? Thank all.
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1answer
13 views

Simplification of a large sum obtained from the 1-D wave equation

I have acquired the sum below through Fourier, and was wondering if there was anyway to simplify it, since it is large and ugly. $$\sum \limits_{n=1}^\infty \frac{-2K_1}{n\pi} ...
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1answer
24 views

sinc in 2d: how to interprete this in spatial domain?

The following two images are the ideal low pass filter in the frequency domain. As you can see, the origin (low frequency component), can pass through this filter while the high frequency are blocked. ...
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1answer
28 views

Setting up my Fourier series for $B_n$

Related but not necessary to know: here Looking at the temperature distribution in an infinitely long cylinder of metal with insulated sides and initial temperature distribution $f(x)= ...
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27 views

Solve this integral with transformations

I have the integral $$ I= \int_0^\infty\frac{w^2dw}{(1+w^2)^4}$$ and I think I can solve it using the fourier transform and Plancherels formula. I can write Plancherels as: $$\int_{-\infty}^\infty ...
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1answer
27 views

Prove that $\hat{f}(n)={\frac{2}{\pi(1-4n^2)}}$, given that $f(x)=|sin(\pi x)|$

Prove that $$\hat{f}(n)={\frac{2}{\pi(1-4n^2)}},\ given\ thatf(x)=|sin(\pi x)||,\int_{0}^{1}sin\pi(x)dx={\frac{2}\pi}\\where\ \hat{f}(x)=\int_{0}^{1}f(x)e(-nx)dx. \ Use\ the\ fact\ ...
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1answer
30 views

Writing a function in terms of the rect and delta functions.

Say I have a function that is equal to 1 at two unit area squares. One is centered at $(-3,0)$ and the other at $(3,0)$. I am trying to find a formula for this function using only the rect function ...
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17 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
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1answer
26 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
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1answer
43 views

Integral for $\frac{x}{x^2+1}cosx$

When computing Fourier transformation I came across these integral: $$ \int_{\Bbb R}\frac{x \cos x}{1+x^2}\;dx\text{ or } \int_{\Bbb R}\frac{x \sin x}{1+x^2}\;dx $$ Can anyone give me some hints on ...
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0answers
18 views

Inverse Fourier Transform of $\cos(c\omega t)$ and $\sin(c\omega t)$

I'm just needing a bit of help to understand the derivation of the inverse fourier transform of $\cos(c\omega t)$ and $\sin(c\omega t)$, in deriving D'Alembert's solution to the wave equation. I get ...
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1answer
38 views

Is it always the case that lower frequencies contribute the most in a Fourier series?

Is it always the case that lower frequencies contribute the most in a Fourier series? Or to put it in other words, in the equation: $$f(t)=a_0+\sum^\infty_{m=1} a_m\cos \left(\frac{2\pi mt}{T}\right) ...
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1answer
74 views

Is there a complete orthornomal basis of a Hilbert space which takes positive values on a discrete set?

Is there a complete orthonormal basis $\{f_n\}$ (of continuous functions) of the Hilbert space of square integrable functions on $[0,\,\infty)$ for which there exists a countable set $S\subset ...
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1answer
25 views

zero distribution of the Fourier kernel $\Phi(u)$ for Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
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97 views

Taking the Fourier transform of a Hankel function

Considering the following inverse Fourier transform $$ f(t) = -\alpha \int_{-\infty}^{\infty} F(\omega)H_0^{(2)}(k(\omega) \beta) \exp(+j\omega t) d\omega$$ where $F$ is an arbitrary function and ...
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23 views

Fourier transform questions [closed]

I have three fairly simple Fourier Transform questions that I need answers to from an independent source in order to prove that my answers are correct. Here are the questions: 1) Find the Fourier ...
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1answer
44 views

Is there a way to relate prime numbers and the fourier transform

According to what I know about Fourier transforms, any continuous periodic signal can be represented as a combination of sine and cosine functions. To me, this looks analogous to the "Fundamental ...
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1answer
14 views

inverse fourier transform of exponencial

Show that $F^{-1}(e^{-|x|}) =(\sqrt{2}/\sqrt{\pi})*1/(1+x^2)$ on $\mathbb R$. $F^{-1}$ is the inverse Fourier transform. Any help? how do you solve the integrals?
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Fourier Transform Identity?

$f(x) \in \mathbb{R}$ and $g(x) \in \mathbb{R}$ $$\int\int \mathop{dx \, dy} f(x)f(y)g(x-y) = \int dk \, \left| \tilde{f}(k)\right|^2\tilde{V(k)} $$ All integrals are over all space. Is this true? ...
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1answer
30 views

Half-Fourier transform, relation to Delta function

so the Fourier transform of the Kronecker Delta function is (up to sign conventions / normalisation) $$\int_{-\infty}^\infty dt\; e^{i t \omega} = \delta(\omega).$$ Can one say anything about the ...
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1answer
37 views

Solution of the integral $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{3xy}{(x^{2}+y^{2}+z^{2})^{5/2})}e^{i(k_{x}x+k_{y}y)} dx dy$

I'm trying to solve the following integral: $$\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}\frac{3xy}{(x^{2}+y^{2}+z^{2})^{5/2})}e^{i(k_{x}x+k_{y}y)} dx dy$$ i'm using the solution ...
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0answers
22 views

complex integral with non integer power

I want to calculate this integral ...
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16 views

What is the best estimation for the following?

Let $f$ be 1-periodic and $f\in L_{p}[0,1]$ where $p>1.$ Let $D_{n}$ $n=0,1,2,..$ be the dyadic partition of $[0,1].$ Consider $$ F_{n}(x)=\frac{1}{|I^{n}_{j}|}\int_{I^{n}_{j}}f(t)dt, ...
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22 views

Obtaining generating function via Fourier transform

Series coefficient for a function can be obtained via Fourier transform: $$f^{(s)}(0)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^s \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, d\omega$$ ...
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59 views

Convolution of two Gaussians or two sinc functions using direct integration

I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions ...
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28 views

Estimation of trignometric polynomial and lipshitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
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Decay of Holder Alpha continuous functions. [closed]

We know that $f\in C^\alpha([0,1])$ implies that $\widehat{f}(n)=O(|n|^\alpha)$. The exercise is show that the exponent in this decay estimate cannot be improved by showing that the function belongs ...
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20 views

how does the following parameters affect the peak ( radius ,theta) of the its frequency spectrum?

The following is the similar magnified picture to the second image that it indicates the parameter i am going to talk about. Suppose i am going to fourier transform (DFT) of the following picture ...
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3answers
100 views

Solution of Definite integral:$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}e^{i(k_{1}x+k_{2}y)}dxdy$

I'm trying to evaluate the following two dimensional integral: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}e^{i(k_{1}x+k_{2}y)}dxdy$ The paper that i'm ...
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35 views

Fourier transform of exponent?

Mathematica fails to find a Fourier transform of exponent. Yet according to this page $$\mathcal{F}[e^{2\pi iat}]=\delta(t-a)$$ and via substitution, $$\mathcal{F}[e^{at}]=\delta\left(t-\frac ...
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2answers
35 views

Two different results of Fourier Transform $xe^{-x}$

I have a function $f$ defined by $$f(x)=\begin{cases} xe^{-x} \textrm{ if } x>0,\\ 0,\textrm{otherwise}. \end{cases}$$ I wish to know the Fourier transform of $f$, i.e, $${\cal ...
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1answer
19 views

Can piecewise $C^{1}$ on $[a,b]$ imply Lipschitz continuity

I saw a statement that if $f$ is continuous,$2\pi$-periodic function which is $C^{1}$ piecewisely on $[-\pi,\pi]$, then its Fourier series converges uniformly to $f$ on $[-\pi,\pi]$. I was wondering ...
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22 views

I need help resolving my problem with DFT

I've been working to understand DFT and my results are not what I would expect. For clarity, I'm using C for T&E and my question isn't C related. My problem is in the DFT and my understanding of ...