All Questions
11 questions
6
votes
1
answer
859
views
How many Fourier coefficients vanish?
Let $G$ be a compact abelian connected metric group with Haar measure $\mu$ and let $f\colon G\to S^1$(=unit circle in $\mathbb{C}$) be a continuous function (not necessarily a group homomorphism) ...
- 3,031
4
votes
1
answer
248
views
Fourier coefficients of Selberg polynomials
In Montgomery's "Ten Lectures on the Interface Between Number Theory and Harmonic Analysis" a bound for the Fourier coefficients of the Selberg polynomial $S^+_K$ is obtained by using what ...
- 3,625
2
votes
0
answers
152
views
Non-commutative harmonic analysis on the discrete Heisenberg group
Question: Is there a linear map $\mathcal F$ from the Hilbert space of $\ell^2$ functions on the discrete Heisenberg group to some Hilbert space of functions $ L^2(\bigcup \{\Omega_n\}) $, such that:...
- 9,501
12
votes
1
answer
457
views
Is there a physical/geometric proof for L^2 boundedness of Bourgain's maximal function along the squares?
One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function ...
- 679
1
vote
0
answers
163
views
Is this averaged exponential sum over primes small infinitely often?
Do there exist infinitely many positive integers $N$ such that
$$\sum_{\substack{N/2 \leq q \leq N \\ a/q \notin \mathbb{Z}}} \left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^...
- 217
7
votes
1
answer
488
views
Examples of the large sieve inequality where a constant larger than 1 is needed
Let $S(x) = \sum_{n=0}^{N-1} a_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that
$$ \sum_{r=1}^R |S(x_r)|^2 ...
- 13k
5
votes
2
answers
476
views
Fourier support condition in the paper 'A study guide for the $l^2$ decoupling theorem'
I'm currently reading Bourgain and Demeter's study guide for the $l^2$ decoupling theorem (https://arxiv.org/pdf/1604.06032.pdf). I have some trouble with understanding the proof of Proposition 8.4.
...
- 61
4
votes
1
answer
532
views
An Exponential Sum Restricted to Primes
Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$.
Is the following estimate true, and, if so, how can it be proved?
\[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \...
- 217
6
votes
0
answers
486
views
On Fourier coefficients of nonnegative function
Let $N$ be nonnegative integer, $F(x)$ be a nonnegative real
Lebesgue integrable function defined on $[0,1]$. Suppose that all Fourier coefficients $c(\lambda)=\int_0^1F(x)e^{-2\pi i \lambda ...
- 12.3k
9
votes
1
answer
912
views
The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?
I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a ...
- 5,169
12
votes
2
answers
1k
views
Fourier transform of the critical line of zeta?
This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
- 4,897