# Questions tagged [trigonometric-polynomials]

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18
questions with no upvoted or accepted answers

**14**

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**0**answers

495 views

### Precise form of the mean motion theorem

Consider an exponential polynomial
$$f(t)=\sum_{k=1}^na_k\exp(i\lambda_kt),$$
where $a_k$ are complex and $\lambda_k, t$ real. The usual form of the Mean Motion Theorem says that the limit
$$\lim_{t\...

**10**

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**0**answers

494 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 ...

**6**

votes

**0**answers

278 views

### 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 ...

**5**

votes

**0**answers

111 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}...

**5**

votes

**0**answers

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&...

**3**

votes

**0**answers

219 views

### 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 $\int_{-\pi}^\pi F_N(t) dt$ ...

**3**

votes

**0**answers

185 views

### Reverse Markov-Bernstein inequality for trigonometric polynomials

Let $r(t)$ be a real trigonometric polynomial of degree $n>1$. Assume it has zero at $t=0$ of multiplicity $k>0$. What can be said about the lower bound of the constant $c(k,n)$ such that
$$
\...

**3**

votes

**0**answers

105 views

### 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 ...

**3**

votes

**0**answers

283 views

### Prove the following trigonometric inequality

Prove that $$f(x, y) \equiv \arccos\left(\frac{x-y}{K}\right) - \arccos\left(\frac{x-y}{K}+y\right) - \frac{y}{x}\arccos(1-y^2) \ge 0$$
with the constraints:
$K\ge 2$ is an integer,
$g(x, y) = (K-1)...

**3**

votes

**0**answers

132 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^{...

**2**

votes

**0**answers

100 views

### 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 $b_0$). ...

**2**

votes

**0**answers

789 views

### Is there an infinite product like this for $\cos x$?

There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...

**2**

votes

**0**answers

95 views

### Octahedron and System of trigonometric equations

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=...

**1**

vote

**0**answers

72 views

### 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 ...

**1**

vote

**0**answers

99 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&...

**0**

votes

**0**answers

37 views

### 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 $K_N(y)=\sum_{|n|<N}\frac{N-|n|}{N}e^{iny}$?

**0**

votes

**0**answers

118 views

### What numbers (irrational) in radicals are expressible as trigonometric rational fraction with only rational multiplies of $\pi$?

What irrational expressions $A$ with radicals can be expressed as trigonometric rational fraction (not a series) with only rational multiplies of $\pi$.
Example:
$ \frac{1}{\sqrt5} = \frac{\sin\frac{\...