All Questions
8 questions
10
votes
0
answers
512
views
Montgomery's conjecture and lower bound on certain Fourier transform.
Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
- 26k
9
votes
2
answers
483
views
Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?
(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem)
Let $f:\mathbb{R}\to [0,\infty)$ be such that
(a) $\int_{\mathbb{R}} f(x) dx = 1$,
(b) $\...
- 20.2k
6
votes
1
answer
679
views
Fourier optimization problem related to the Prime Number Theorem
Let $\kappa>0$ be given. What is the function
$f:\mathbb{R}\to [0,\infty)$ with $\int_\mathbb{R} f(x) dx = 1$ such that
$$\int_\mathbb{R} |x| f(x) dx + \kappa \int_{|t|\geq T}\left| \frac{\widehat{...
- 20.2k
5
votes
2
answers
245
views
An optimization problem: $\Phi(0)$, $\widehat{\Phi}(0)$, $\Phi$ a majorant
(This is a problem that arose from my own answer to Mean value theorem for Dirichlet series - optimize? )
Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such ...
- 20.2k
5
votes
0
answers
326
views
Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
\frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
- 1,199
2
votes
0
answers
113
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
2
votes
0
answers
187
views
Three optimization problems of uncertainty principle/Paley-Wiener type
Let $\phi:\mathbb{R}\to\mathbb{R}$ be an even function with support on $[-1,1]$. Assume that it is in $L^1\cap L^2$ and that its Fourier transform is also in $L^1\cap L^2$. Assume as well that $|\phi|...
- 20.2k
2
votes
0
answers
79
views
For $\Phi$ a majorant of $1_{[-1/2,1/2]}$, how small can the total variation of $\widehat\Phi$ be?
Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such that $\Phi(t)\geq 1$ for $|t|\leq 1/2$. Assume furthermore that $\Phi$ and $\widehat\Phi$ are both in $L^1\...
- 20.2k