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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Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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Strichartz Estimate with Fourier Transform

Let $f$ be a Schwartz function. Prove that, whenever $2\le r < \infty,$ $$\| e^{it \Delta} f\|_{L^{3r}(\mathbb{R}^2_{xt})} \le c \| \widehat{f}\|_{r'},$$ Where $1/r + 1/r' = 1.$ My Attempt My ...
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How to get this result of integral?

Statement \begin{equation} \int_{\mathbb R} \exp \left( -2\pi (\frac{x}{\sqrt{2}})^2 \right) \exp\left( -i2 \pi \frac {x}{\sqrt{2}} \cdot f \right) dx = \exp \left( -\pi f^{2} / 2 \right) ...
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Fourier transform of $H(-t)e^{5t}$

i have to calculate the Fourier transform in the title. My professor says the result is $\frac{1}{5-2\pi i f}$. I start from $H(t)e^{\alpha t}$, and i calculate the transform $H(t)e^{5t}\rightarrow ...
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Can you kindly explain me in detail this Fourier transform?

I've this function to transform not using the general formula, but just substituting the known transform (i.e. $\text{rect}(t)\rightarrow \text{sinc}(f)$): $\frac{\sin(6\pi t)}{t}$ I know the ...
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
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32 views

Fourier Transform of $f : x \mapsto \frac{24(5-5x-30x^2-10x^3+5x^4+x^5)}{(2+x+x^2)^5}$

I need of the Fourier Transform of $f$, but I can't solve this. I try to use the integrate and Fourier proprietates but no sucessufull. Help me please $$\ f(x) ...
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Calculating Fast Fourier Transform from given set of data

I am trying to calculate the Fast Fourier Transform numerically from the given data : Given: f0 f1 f2 f3 f4 f5 f6 f7 1 2 3 4 4 3 2 1 I have to find the ...
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18 views

Tempered representatives of a special class of distributions

Suppose that a distribution $R\in D'(\Bbb R)$ satisfies the following estimation for an independent constant $c$: $$\forall \phi\in D(\Bbb R)\quad |\langle R,\phi\rangle|\le c\|\phi, \,L^1(\Bbb ...
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Estimate of a Fourier Multiplier Operator

Let $m_t (\xi) = \cos (2\pi |\xi| t).$ Define the operators, for $t>0,$ $$ T_t f = ( m_t \widehat{f} )^{\vee}.$$ It is asked to prove that, whenever $f$ is sufficiently regular, $$ \| T_t ...
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Inverse Fourier Transform by using convolution theorem.

Inverse Fourier Transform of: $$\mathfrak{F}^{-1} \left \{ e^{-\frac{x^2}{2}}{\frac{sinx}{x}} \right \} $$ by using convolution theorem. Since Fourier Transform convolution turns into ...
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How to use Fourier's transform to solve differential equation

I have to solve following problem: $$ u_t(t,x) = \Delta u(t,x) $$ $$ u(0,x) = f(x) $$ I've started: $$ \frac{\delta}{\delta t} F(u(t,\xi))=F(u_t(t,\xi))=F(\Delta u(t,x))$$ and here I've stoped, ...
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1answer
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Fourier transform of $\frac{\sin(6\pi t)}{t}$

I have to calculate the fourier transform of this function in time domain: $\frac{\sin(6\pi t)}{t}$. First I tough to use the definition of $\operatorname{sinc}$ function as ...
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Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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Mathematical Expression for a Fourier Transform $s(T)$

$S(f)$ is the Fourier transform of a non-periodic signal, $s(t)$. $S(f)$ is given by: $S(f) = 1,$ for $−1/2 ≤ f ≤ 1/2$ and $0$ otherwise. What would be a mathematical expression for $s(t)$?
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Use the Fourier transform to find value of definite integral from negative infinity to infinity

Find the value of $\int_{-\infty}^{\infty} f(x) dx$, where $f(x)=sin(x)/(x^3+x)$. How do I go about solving this? I have tried to expand the sine part into complex exponentials to try and resemble ...
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28 views

Solve problem using Fourier's transform

I have a few problems which need to be solved using Fourier's transform. My problem is that I don't know how should I start this type of exercise (I just begin learning differentional equations). ...
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Fourier Transform of $f(t)=4te^{-t^2}$

I am trying to find Fourier Transform of: $$f(t)=4te^{-t^2}$$. I found in MatLab that $\mathfrak{F}\left \{ f(t) \right \}=i\sqrt{2}e^{- \frac{w^2}{4}}w$ .So is this possible to come to same result ...
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Confusion of one definition in Fourier analysis

The symbol occurs on Page 22 of Bahouri's book Fourier analysis and nonlinear differential equations. As defined there, $$f(D)a:=\mathcal{F}^{-1}\{f\mathcal{F}a\}.$$ The question comes from the ...
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Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft. Greetings I'm trying to rebuild a signal from the frequency, amplitude, and phase obtained after I do ...
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Convolution of distributions.

$\newcommand{\supp}{\operatorname{supp}}$ We are given with distributions $f,g \in D'(\Bbb R)$. If $\supp f\subset (-\infty,a)$ and $\supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined ...
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How do you take the discrete Fourier transform (DFT) of a parallelogram or a Bravais lattice in general?

I'm working on implementing a method that extracts the corresponding wallpaper group given a gray-scale image/pattern. But to do so, I need to take the DFT of a unit cell in the image which, in the ...
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Fourier transform of windowed complex exponential

I have a function on the form $$f(x) = g'(x)*e^{i\pi g(x)}.$$ Where $g'(x)$ is a window function with support in the range $-R \ldots R$. I want to find the fourier transform $\mathcal F(\omega)$ ...
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What is an angle in fractional fourier transforms?

I would like to know the geometrical interpretation of an angle in fractional Fourier transforms. Is this a rotation of time-frequency plane or rotation of the signal?
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Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
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1answer
25 views

Extension of Fourier transform to complex analytic function

Let $f(x) \in L^1(\Bbb{R})$ have compact support, say $\operatorname{supp}(f) = [-R,R]$. We have the Fourier transform $$\hat{f}(\xi) := \int_{\Bbb{R}} e^{-ix\xi} f(x) dx = \int_{-R}^R e^{-ix \xi} ...
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fourier transform for pde equation

I was solving the pde using fourier transform: $u_{tt}-u_{xx}+m^2u=0$ with initial values $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$. I have received the answer $$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt ...
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Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
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Should I trust Mathematica or numerous other sources on this Fourier transform

Assume $a>0$ So Mathematica claims $$F\{e^{-a|t|}\}(\omega) = \frac{a\sqrt{\frac{2}{\pi}}}{a^2+\omega^2}$$ However, I've read about another transform pair (page 3): $$F\{e^{-a|t|}\}(\omega) = ...
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Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
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When are Fourier spaces included in each other?

Given the Fourier spaces $V(N_1, T_1)$ and $V(N_2,T_2)$, what necessary and sufficient conditions are required in order to have $V(N_1, T_1)\subset V(N_2,T_2)$? I know that if $V(N_1, T_1)$ is ...
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Verifying Convolution Identities

Note: I don't yet have a solution to my main issue yet which I have elaborated on in the edit. Further attention is deeply appreciated. :> $\bf{\text{Original Question}}$: Let $G$ be a locally ...
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Explanation of this integral

$$\int_{-\infty}^{\infty} e^{-\frac{i}{\hbar}(p-p')x} dx = 2\pi\hbar\delta(p-p')$$ I don't quite understand how this integration leads to the right hand side. Any explanation is appreciated.
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Solution of boundary value problem using Fourier series

I want to solve the following PDE using Fourier series. $u(x,y): \Omega \to \mathbb{R}$, $\Omega=(0,\pi)\times (0,2\pi)$ $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$ $u_{xx}$ and $u_{yy}$ are second ...
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Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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Example of multidimensional Fourier transform

Please, take the function, for example $\sin(xy)+\cos(xz)$ from dimension 3 to $R$, and give me a multidimensional Fourier transform for it. I'll be also thankful for general multidimensional Fourier ...
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Solving integral equation with Fourier transform?

I'm trying to solve the following integral equation using Fourier transforms: $$u(t)+ \int_{-\infty}^{t} e^{\tau-t} u(\tau)\,d\tau=e^{-2|\tau|}$$ I tried to transform both sides of the equation using ...
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1answer
35 views

How to properly shift in frequency domain an already shifted function in time domain?

I would like to shift in frequency domain the following function: $f(t)=\frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{(t-t_0)^2}{2\sigma^2})$. As usual, frequency shift will introduce a new term ...
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Convolution Integral to Evaluate Fourier Transform

According to Mathematica with Fourier transform convention $$\widehat{f}(\xi)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{i\pi x}dx$$ The Fourier transform of the function $f(x):=|x|^{-1/2}e^{-|x|}$ ...
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Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq ...
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When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
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What happens with Fourier transform of a Gaussian function?

I am facing the following problem: I have a Gaussian function of time with the following form $f(t)=\frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{t^2}{2\sigma_2})$. I would like to make a shift both in time ...
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Does this transformation(harmonic analysis) exist?

Assuming that there are two disjoint sets(A and B) of high dimensional $N^d$ integers. Each point $V$ can be expressed by a periodic function: $f(t)=v_1*cos(t) + v_2*sin(2t) + ...
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How to identify a process via its Karhunen-Loeve expansion?

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi ...
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fourier transform of $f(x) = x^2+\frac{1}{1+2x^4}$

I really have no thought on this. I can't seem to use residue thm., nor could I find a inverse transform for it. by some Fourier Calculator I know it's solvable but how?
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Using the discrete fourier transform to approximate the regular fourier transform

This may be an elementary question, but I'm not sure how DFT/FFT is used to approximate regular Fourier transforms. Consider the radial distribution function $g(r)$. The structure factor is defined ...
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Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$: $T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial ...
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What's the formulation of N-point radix-N for NTT

We can write the formulation for the buttlerfly function applied in FFT as \begin{align*}y_0 &= x_0 + x_1,\\ y_1 &= x_0 - x_1. \end{align*} As seen here. For FFT (Fast Fourier Transform) we ...
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Reading the properties of Discrete Fourier Transformation from the given figure.

Can anybody please help me read the properties of Discrete Fourier Transformation from the given figure. Here is the image link Thank you guys, appreciate you help.