# Questions tagged [fourier-transform]

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### Discretizing a differential operator which is a function of the derivative operator

Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$. As an example, let $p(x)=x^{2}+2x$. Then ...
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### Proof for $\Phi(t)$ is strictly decreasing for $t>0$ in Riemann's zeta function

I am looking for reference for proof that $\Phi(t)$ is strictly decreasing for $t>0$ and the first derivative of $\Phi(t)$ is negative for $t>0$ (see Page 5 in Conrey's article below) Conrey ...
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### 2-Wasserstein metric on convolution of probability distributions

I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
137 views

### An oscillatory integral

Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates? \begin{...
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### Hyperplanes which equalize the Radon transforms of two distributions

Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
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### What is the 3-dimensional Fourier transform of $1/k^4$?

In electrostatics, we often encounter the following 3-dimensional integral: \begin{equation} V=\int d^{3}\vec{k}\,\dfrac{e^{i\vec{k}.\vec{r}}}{|\vec{k}|^{2}} \end{equation} which yields the Coulomb ...
1 vote
52 views

### Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...
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1 vote
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### On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)

Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
75 views

### Discrete Fourier transform up to a diagonal matrix

Let $N$ be a natural number and $w=e^{\frac{2\pi i}{N}}$ and $u=e^{\frac{\pi i}{N}}$. Let's consider the diagonal matrix $D=\operatorname{diag}(1,u,u^2,\cdots,u^{N-1})$. It is well known that the set ...
1 vote
### How to prove the Fourier transform of $e^{-x^p}$ is positive [duplicate]
I wonder how to prove that $$\int_0^\infty\exp(-x^p)\cos(tx)\,dt\geq 0, \quad \frac{1}{2}<p<1.$$ This conclusion is used in the answer to another question here Looking for sufficient conditions ...
I am wondering how to prove the below Fourier transform is non-negative? I did much simulation and it seems to be non-negative. \int_0^\inf (be^{-at^p}-ae^{-bt^p})\cos(tx)dt, 0<a<b, \frac{1}{2}...