Skip to main content

All Questions

Filter by
Sorted by
Tagged with
2 votes
0 answers
109 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{...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
79 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 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|...
H A Helfgott's user avatar
  • 20.2k
2 votes
3 answers
457 views

Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc

I am considering a PhD research topic. I only have a math Bachelor's degree with working experience mostly in teaching and I have been working on a paper. I have deep interest in Fourier Series and ...
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) $\...
H A Helfgott's user avatar
  • 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{...
H A Helfgott's user avatar
  • 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\...
H A Helfgott's user avatar
  • 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 ...
H A Helfgott's user avatar
  • 20.2k
4 votes
1 answer
295 views

Fourier coeffients of Cantor measure

For $0<\theta<\frac{1}{2}$, denote by $\mu_\theta$ the uniform Cantor measure with dissection ratio $\theta$. It is not hard to show that the Fourier–Stieltjes transform of $\mu_\theta$ is $$ \...
user119197's user avatar
5 votes
1 answer
271 views

For which sets $E\subset \mathbb{Z}_n$ is $\widehat{1(E)}$ nonzero everywhere?

I apologise if this is well-known or straightforward. Define the Fourier transform of the characteristic function of a subset $E\subseteq\mathbb{Z}_n$ by $$ \widehat{1_E}(k)=\sum_{a \in E} \exp(-2 \...
kodlu's user avatar
  • 10.4k
2 votes
0 answers
224 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n =1}^{\...
Bertrand's user avatar
  • 1,199
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)...
Bertrand's user avatar
  • 1,199
5 votes
2 answers
822 views

Is there a Poisson Summation formula for imprimitive Dirichlet characters?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ? For a primitive Dirichlet character $\chi$ we have: ...
Bertrand's user avatar
  • 1,199
5 votes
3 answers
2k views

Extension of Poisson Summation formula

Under the condition f continuous, integrable and: $|f(t)| + |\hat{f}(t)| \le C (1+|t|)^{-1-a}$ (with a>0) we have the twisted Poisson formula that holds (where $\chi(n)$ is a primitive Dirichlet ...
Bertrand's user avatar
  • 1,199