All Questions
Tagged with trigonometric-sums trigonometric-polynomials
9 questions
2
votes
1
answer
230
views
$L_p$ norms of $0-1$ exponential sums
Consider $f_n(t)=\sum_{i=1}^{n}e^{ik_{i}t}$ with all $k_i$ some distinct integers for $t\in [-\pi,\pi)$. For $p>2$ I am interested in the maximum possible value of $$||f_n||_p,$$
where $f_n$ runs ...
- 2,288
8
votes
2
answers
951
views
Better trigonometrical inequalities for $\zeta(s)$?
The inequality $$3 + 4 \cos \theta + \cos 2 \theta \geq 0$$ plays a key role in the proof of the classical zero-free region of the Riemann zeta function. Are there other inequalities of the form
$$\...
- 20.2k
4
votes
0
answers
289
views
A uniform Riemann sum approximation of the integral of the Fejer kernels
Let $F_N(t)$ denote the Fejer kernel
$$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$
Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...
- 136
2
votes
0
answers
112
views
Lower $L^1$ norm estimates of null average trigonometric polynomials depending on the order of the polynomial
Let $p(x)=\sum_{k=1}^m [a_k\cos(n_kx)+b_k\sin(n_kx)]$ be a null average trigonometric polynomial (null average means that is $\int_\mathbb T p =0$ or, equivalently, there are no $a_0$ and $b_0$). ...
- 151
7
votes
2
answers
853
views
Closed formula for sine powers
I am looking for a closed formula for the expressions
$$ \sum_{k=1}^{n-1} \sin\left(\pi \frac{k}{n}\right)^m,$$
with $n \in \mathbb{N}$ and $m \in \mathbb{N}$ odd.
Playing with these sums a bit, I ...
- 9,853
6
votes
1
answer
399
views
Eigenvalues of partial Hankel matrices
I was wondering if there are closed formulas for the singularvalues of a partial Hankel matrix (by partial I mean $\ell<n$)
\begin{align*}
H=
\begin{bmatrix}
c_1 & c_2 & \ldots & c_\...
- 363
2
votes
1
answer
682
views
Maximal minimum for a sum of two (or more) cosines
Please prove (or disprove, and give the correct answer):
$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right]
$$
In other words, find $r \geq 1$, ...
- 968
3
votes
0
answers
152
views
Bounding expected value of maximum of dot product with random chirp
Let $\mathbf{x}\in\mathbb{C}^n$ with $\|\mathbf{x}\|=1$ with $n<\frac{N}{2}$. I am interested in a bound of the form
\begin{equation*}
\mathbb{E}\Big\{\max_{k\in\{1,2,\ldots,n\}}\Big|\sum_{a=1}^ne^{...
- 859
5
votes
0
answers
117
views
Concentration of weighted random chirp
I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have
\begin{align*}
\mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}...
- 859