# Questions tagged [trigonometric-polynomials]

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### 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 ...
1 vote
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### (update) Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?

Problem: Given three positive integers $0 < n_1 < n_2 < n_3$ such that $$n_1 + n_2 \ne n_3, \quad n_2 \ne 2n_1, \quad n_3 \ne 2n_1, \quad n_3 \ne 2n_2,$$ is there always a real number $x$ ...
384 views

### Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?

Problem: Given three positive integers $0 < n_1 < n_2 < n_3$. Is there always a real number $x$ such that $$\cos n_1 x + \cos n_2 x + \cos n_3 x < -2?$$
132 views

1 vote
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### Solutions of equation $\sin \pi x_1\sin \pi x_2=\sin \pi x_3\sin \pi x_4$ [closed]

I am interested in finding all the solution $(x_1,x_2,x_3,x_4)\in \mathbb{R}^4$ of equations: $$\sin \pi x_1\sin\pi x_2=\sin \pi x_3\sin\pi x_4.$$ I have found out a paper: Rational products of sines ...
481 views

### Curious identity between the two kinds of Chebyshev polynomials

I have found, by accident, an identity that relates a sum of Chebyshev polynomials of the first kind to a Chebyshev polynomial of the second kind. It goes as follows: Given an integer partition of $n$...
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### 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 $b_0$). ...
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### Approximating $1_I$, $I\subset \lbrack 0,1\rbrack$, by trigonometric polynomials

Let $I$ be a subinterval of $\lbrack 0,1\rbrack$. How well can one hope to approximate it in the $L_1$-norm (on $\mathbb{R}/\mathbb{Z}$) by a trigonometric polynomial of degree $\leq R$, that is, a ...
787 views

### An extremal problem related either to an uncertainty principle on the circle, or else to the prime number theorem

Consider for $X = 1,2, \ldots$ the set $\mathcal{S}_X$ of trigonometric polynomials $f(t) := \sum_{|k| \leq X} c_k e^{2\pi i kt}$ on the circle $\mathbb{T} := \mathbb{R}/\mathbb{Z}$ of degree $\leq X$ ...
1 vote
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### Is there a law of cosine for n-dimensional hyperbolic simplex

We know that, given an n-dimensional Euclidean simplex, for all $1\leq i,j,k,l\leq n+1$, we have(law of sines)$$\frac{A_i A_j}{A_k A_l}=\frac{c_{ij}}{c_{kl}}$$(from Elementary Formulas for a ...
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### cosine of rational multiples of Pi take values of equal difference

In my physics research I came across a mathematical proposition (translated into the mathematical language from the physical problem) that I feel to be true, and would like to prove it: Proposition: ...
832 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 ...
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### 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_\...
101 views

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ $$\sum_{k=... 2 votes 1 answer 517 views ### Maximal minimum for a sum of two (or more) cosines Please prove (or disprove, and give the correct answer):$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right] $$In other words, find r \geq 1, ... 9 votes 1 answer 507 views ### L^1 norm of exponential sum of n^2 x What is the asymptotic order of$$ \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 1 answer 138 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 \{... 10 votes 0 answers 536 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 ... 3 votes 0 answers 147 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^{... 5 votes 0 answers 116 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}... 7 votes 1 answer 389 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( a_{kj}\sin(jt)+...
1 vote
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 ...
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, \$ 0<\alpha&...