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1
vote
1answer
118 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 $r \geq 1$, ...
9
votes
1answer
293 views

$L^1$ norm of exponential sum of $n^2 x$

What is the asymptotic order of $$ \int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx $$ as $N \to \infty$. This should be known, but I cannot find it in the literature.
4
votes
1answer
78 views

Estimate self crossings of a curve parameterized by a trigonometric polynomial

Given z on the unit circle, let $P(z)= \sum\limits_{k=-D}^D p_k z^k $. Can one estimate the number of self crossings of the following curve with an analytic expression in terms of the coefficients ...
9
votes
0answers
245 views

Reciprocal polynomials with roots off the unit circle

A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...
1
vote
0answers
78 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*} ...
3
votes
0answers
85 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*} ...
7
votes
1answer
270 views

A conjecture about the measure estimates of a trigonometric polynomial

Formulation of the Conjecture Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( ...
1
vote
0answers
81 views

Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...
1
vote
0answers
237 views

Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$ where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...
5
votes
0answers
206 views

trigonometric polynomial

Can anyone tell me the following statement is true or not? Thank you. There are two polynomials: \begin{align} p(r,\theta) &=\sin(n_0\theta) + \sum_{j=1}^{\ell}a_j r^{n_j}\sin(n_j\theta), \quad ...
2
votes
1answer
364 views

Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...
8
votes
1answer
340 views

certain trigonometric homeomorphisms

Are there any simple characterizations of rational functions $f(x,y)$ with real coefficients such that $\theta\mapsto f(\cos\theta,\sin\theta)$ is a homeomorphism from $\mathbb R\bmod 2\pi$ to ...
2
votes
1answer
157 views

Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form \begin{equation*} f(t)=\sum_{k=-n}^n c_k ...
2
votes
1answer
246 views

bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$ \begin{equation*} f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})} ...