# Questions tagged [trigonometric-polynomials]

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### Majorizing $|\{\alpha\}-1/2|$ by trigonometric polynomials

Let $f(\alpha) = |\{\alpha\}-1/2|$. What is the trigonometric polynomial $F_N$ of degree $N$ (i.e., a linear combination $\sum_{n=-N}^N a_n e(\alpha n)$, $a_n\in \mathbb{C}$, where $e(r)= e^{2\pi i r}$...
1 vote
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### What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]

sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there: How can $\sqrt{2x^2}$ become $4x^2$ ?
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### Polynomial satisfied by $\cos^n(t)$ and $\sin^n(t)$

For any even $n$, there should be a polynomial $f(x,y)$ that vanishes on the points $(\cos^n(t),\sin^n(t))$ for all $t$ (since it is the image of the projective variety $x^2+y^2 = 1$ under the $n/2$th-...
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### Optimization problem on trigonometric polynomials

I would like to maximize $$\int_0^{2\pi} \frac{(f'(x))^2}{f(x)}dx$$ subject to $f(x)\leq 1$ for all $x$ over the space of nonnegative trigonometric polynomials of degree smaller or equal to $n$. ...
1 vote
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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 ...
1 vote
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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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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 $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( a_{kj}\sin(jt)+... 1 vote 0 answers 100 views ### Estimating decay of certain trigonometric polynomials For p=0,1,2,\dots and n=0,1,2,\dots,, let f_{n,p}(z)=\sum_{k=0}^n k^p z^k be a sequence of polynomials. Restricted to the unit circle, the functions g_{n,p}(t):=f_{n,p}(e^{it}) are ... 1 vote 0 answers 286 views ### Proof of an inequality for a linear combination of three trigonometric functions Given a function$$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, 0<\alpha&... 5 votes 0 answers 254 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&... 3 votes 1 answer 781 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$. Define the upper uniform density (... 8 votes 1 answer 429 views ### certain trigonometric homeomorphisms Are there any simple characterizations of rational functions$f(x,y)$with real coefficients such that$\theta\mapsto f(\cos\theta,\sin\theta)$is a homeomorphism from$\mathbb R\bmod 2\pi$to$\...
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})} \...