The trigonometric-polynomials tag has no wiki summary.

**7**

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**0**answers

171 views

### Functions mapping [-1,1] to itself with high linear coefficient

Given a real analytic function that can be written as
$$
f(x)=\sum_{i=1}^\infty \lambda_i x^i,
$$
and a real $1\gg \epsilon >0$, define the functional
$$
\phi_\epsilon(f) := |\lambda_1| - \epsilon ...

**1**

vote

**1**answer

112 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

275 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

77 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 ...

**-1**

votes

**1**answer

170 views

### What is the advantage of the knowledge of jumps for approximating a function with trigonometric polynomials?

Let $f:(a,b)\to\mathbb{R}$ be square integrable, bounded variation and piecewise continuous function. Let the points of jump be $\{x_i/a<x_i<b,i = 1,2,3,...n\}$. The goal is to approximate the ...

**9**

votes

**0**answers

242 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

**0**answers

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

**0**answers

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

**1**answer

266 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

**0**answers

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

**0**answers

235 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

**0**answers

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

**1**answer

354 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

338 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

**1**answer

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

**1**answer

245 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})}
...