# Questions tagged [fourier-transform]

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### Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$V(x) = V_0 \mathbf 1_{[-a,a]}(x),$$ where $\mathbf 1_{[-a,a]}$ denotes the ...
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### Fourier Transform diagonalizes time-invariant convolution operators [closed]

I got the following paragraph from the book "A wavelet tour of signal processing" chapter one, page 2. The Fourier transform is everywhere in physics and mathematics because it diagonalizes ...
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### Integration against a certain Fourier transform

I asked the following question on mathstack but didn't receive any answers. I suspect that this question has a simple answer but I haven't thought about Fourier transforms in a while so am being ...
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### A question about Fourier transform of a function defined by an integral

I have the function: $$G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta},$$ for all $x\in\mathbb{R}^n$ and $k>0$....
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### Fourier transforms exhibiting symmetries about their critical points

Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
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### General Fourier inversion formula (Gil-Pelaez)

Gil-Pelaez (1951) proves the Fourier inversion formula \begin{align*} F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\ &= \frac{1}{2} - \frac{...
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### Failure of Strichartz estimates for the wave equation: elaboration of a counter-example

One can read in Oh - Probabilistic perspectives in nonlinear dispersive PDEs (Proposition 64, p. 60) that there exists a function $F \in L^2_tL^{1}_x (\mathbb{R}_t\times \mathbb{R}^3_x)$ which is ...
I wonder how to solve the Klein-Gordon equation $$\left\{\begin{split}&\partial_t^2u-\Delta u+u=f\\&u(0,x)=u_1(x),\quad \partial_t u(0,x)=u_2(x)\end{split}\right.$$ where $u(t,x)$ defined on $\... 0answers 38 views ### Computing the matrix of first order marginals from low frequency terms of the Fourier Transform Let$S_n$be the symmetric group and$f:S_n \to \mathbb{R}$be a function on$S_n$. We define the Fourier Transform of$f$as the collection of matrices$$\hat{f}_{\lambda} = \sum_{\sigma \in \... 0answers 64 views ### Particular Ehrenpreis factorization for covariance function Let$f:\mathbb{R}^d\to\mathbb{R}$be a smooth compactly supported covariance function of a stationary random fields (hence positive definite). Is there a compactly supported function$g:\mathbb{R}^d\...
Suppose I have a signal $u(x)$ where $x\in[0,2\pi]$ and the signal has following form in Fourier space, \$\hat{u}(\kappa)=\begin{cases} \frac{1+i\tan{(\varphi_u(\kappa))}}{\sqrt{1+\tan^2{(\varphi_u(\...