All Questions
Tagged with fourier-transform mp.mathematical-physics
14 questions
12
votes
1
answer
1k
views
Fourier transform on Minkowski space
Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...
- 64k
6
votes
1
answer
255
views
Is this function Schwartz?
I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it.
Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that
$$
\...
- 428
6
votes
0
answers
159
views
Fourier transformation of a distribution
We have no idea how to tackle the following Fourier transformation of a distribution:
$$
\lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
- 373
5
votes
4
answers
952
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
4
votes
2
answers
1k
views
The Hubbard-Stratonovich transformation
is there a way to extend the Hubbard-Stratonovich transformation
$$e^{\frac{1}{2}Ks^2}=\left(\frac{K}{2\pi}\right)^{1/2}\int_{–\infty}^\infty e^{–Kx^2+Ksx}dx$$
to the case $e^{\frac{1}{2}Ks^p}$ for $...
- 197
4
votes
2
answers
405
views
Fourier transform of a Lorentz invariant generalized function
Consider on $\mathbb{R}^{n+1}$ the indefinite quadratic form defining the Minkowski metric
$$B(p)=(p^0)^2-(p^1)^2-\dots-(p^n)^2.$$
Let $\mu$ be a generalized function on $\mathbb{R}^{n+1}$ which is ...
- 21.8k
2
votes
1
answer
118
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Is the Fourier multiplier $\mathcal F(G(-\hbar^2 \Delta)\psi)(p) = G(|p|^2)\hat \psi(p)$ justified for any real function $G$?
I am confused about a claim asserted in the paper "Higher Order Schrodinger Equations" published in IOP Science. The authors claim that a Fourier multiplier identity
$$
\mathcal F(G(-\hbar^2 ...
- 547
2
votes
0
answers
156
views
About solutions of Klein-Gordon equation
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 $\...
- 71
1
vote
3
answers
307
views
Fourier transform of a generalized function on the plane
Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
$$\frac{1}{x+y^2+i0}?$$
Remark. Equivalent question: consider the Schroedinger equation one the ...
- 21.8k
1
vote
1
answer
573
views
Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$
Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished computational basis $e_i \otimes e_j $ and a group of translations $T_a$ defined by $T_a e_i \otimes e_j = e_{i+a} \...
- 1,054
1
vote
0
answers
74
views
Calculation of a multi-dimensional Fourier transform
I am interested in the following multi-dimensional Fourier transform:
$$
\int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}...
- 373
1
vote
0
answers
127
views
Does convergence of tempered distributions implies convergence in $\mathcal{S}(\mathbb{R}^4,\mathbb{R})/\mathcal{S}_{0}$?
We can define the following symmetric semi-definite positive bi-linear form on
$\mathcal{S}(\mathbb{R}^{4},\mathbb{R})$ with values in $\mathbb{C}$,
\begin{equation}\label{prodintespaciales}
(h_{...
- 424
0
votes
1
answer
410
views
About vertex algebra, mode expansion
A vertex operator is a linear map associating every state to a operator-valued distributions (quantum field) on a algebra curve, which is also called operator-state correspondence.
Chose a local ...
- 787
-2
votes
1
answer
363
views
Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT
In one line: Given an exponentially decaying sine wave $x(t)$, how can we predict the amplitude of the resulting peak in frequency spectrum using discrete Fourier transform.
In nuclear magnetic ...
- 879