# Questions tagged [trigonometric-polynomials]

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### Convolution of Fejer kernel

I have a question about how to prove $$|p(y)|\leq 3\int_{T^2}|p(y-z)|K_N(z)dz$$ where $p(y)=\sum_{|n|<N}a_n e^{iny}$ and $K_N(y)=\sum_{|n|<N}\frac{N-|n|}{N}e^{iny}$?
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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_\...
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### Tight upper bounds on trigonometric polynomials

According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
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### Closed form of $\prod_{k=1}^{n}\left(\cos(kx)-1\right)$

Is there any closed form of $$\prod_{k=1}^{n}\left(\cos(kx)-1\right)?$$ I failed to find references on this problem in the internet.
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### Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?

We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain. Is it possible to estimate ...
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### Is there a nullstellensatz for trigonometric polynomials?

Let $$f(x) = \sum_{j=1}^n c_j e^{2\pi i\alpha_j x}, g(x) = \sum_{k=1}^m d_k e^{2\pi i\beta_k x}$$ be two (quasi-periodic) trigonometric polynomials, where the coefficients $c_j, d_k$ are complex and ...
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### $L_p$ 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 ...
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### Reference request: Gaussian almost periodic functions

Let $X(x),x\in R^d$, be a stationary gaussian process for which the covariance function $E(X(0)X(x))=C(x)$ is "almost periodic". Almost periodic means roughly that $C$ is uniformly approximable by ...
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### Are all trigonometric polynomials from the 3-torus to the 3-sphere homotopically trivial?

I'm looking at maps from the 3-torus $\mathbb{T}^3\simeq (\mathbb{R}/2\pi\mathbb{Z})^3$ to the 3-sphere $\mathbb{S}_3\subset \mathbb{R}^4$. I understand that, according to Hopf theorem, continuous ...
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### 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 $\int_{-\pi}^\pi F_N(t) dt$ ...
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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: ...
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### 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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### 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 ...
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### 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^{...
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### 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}...
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### 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 ...
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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&... 0answers 249 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 r&... 1answer 419 views ### certain trigonometric homeomorphisms Are there any simple characterizations of rational functionsf(x,y)$with real coefficients such that$\theta\mapsto f(\cos\theta,\sin\theta)$is a homeomorphism from$\mathbb R\bmod 2\pi$to$\...
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 e^{2\...
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})} \...