31
votes
Accepted
Is there a nullstellensatz for trigonometric polynomials?
I'm converting the comment above to an answer:
Shapiro made the following conjecture in the paper "The expansion of mean-periodic functions in series of exponentials" Comm. Pure and Appl. ...
- 85.6k
28
votes
Accepted
(update) Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
In principle this problem can be resolved numerically in finite time, by exploiting the dichotomy between structure (small linear relations between the frequencies $n_1,n_2,n_3$) and randomness (...
- 114k
17
votes
Accepted
Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
The answer is negative. If $n_3=n_1+n_2$, then
$$\cos n_1 x + \cos n_2 x + \cos n_3 x\geq -2.$$
Indeed, the left-hand side equals
$$(1+\cos n_1 x)(1+\cos n_2 x)-1-\sin n_1 x\sin n_2 x,$$
where $1+\cos ...
- 105k
17
votes
Accepted
Better trigonometrical inequalities for $\zeta(s)$?
Assuming the $b_i$ are all distinct (or at least non-zero for $i \neq 0$), this is not possible. (Otherwise there are trivial examples, e.g. $1 + 2 \cos(0 \theta)+ \cos(0 \theta) \geq 0$ or $1 + 4 \...
- 114k
16
votes
Closed formula for sine powers
You are looking at the sum of $m$-th powers of the imaginary parts of $n$-th roots of $-1.$ First, note that these can be expressed as polynomials in elementary symmetric functions (of the imaginary ...
- 96.4k
13
votes
Polynomial satisfied by $\cos^n(t)$ and $\sin^n(t)$
I don't know if this is irreducible, but it gives an answer: Let $\zeta$ be a primitive $n/2$-root of unity. Multiply out
$$\prod_{a=1}^{n/2} \prod_{b=1}^{n/2} (\zeta^a x^{2/n} + \zeta^b y^{2/n} - 1 )....
- 156k
11
votes
Accepted
Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots
Let me first start with the other side: does the maximum being small guarantee that the roots are equidistributed? This is indeed the case, and is a beautiful theorem of Erdos and Turan. For a ...
- 43.7k
11
votes
Accepted
An extremal problem related either to an uncertainty principle on the circle, or else to the prime number theorem
A compactness argument shows that for sufficiently large $X$ one has the bound
$$ \sup_{x \in {\mathbb T} \backslash [-1/X,1/X]} |f(x)| \gg \sup_{x \in [-1/X,1/X]} |f(x)|$$
whenever $f$ is a ...
- 114k
11
votes
Accepted
Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?
Since $a,b$ are incommensurable, $(ax,bx)$ is asymptotically equidistributed
in the torus $({\bf R} / 2\pi{\bf Z})^2$.
[One proof is via a continuous version of Weyl's equidistribution criterion:
for ...
- 79.9k
9
votes
Polynomial satisfied by $\cos^n(t)$ and $\sin^n(t)$
You can use polynomial elimination, which is implemented in Macaulay2.
In fact, the parametric equations of your affine curve are $$x=\frac{(2t)^n}{(1+t^2)^n}, \quad y=\frac{(1-t^2)^n}{(1+t^2)^n},$$
...
- 66.3k
8
votes
Accepted
Curious identity between the two kinds of Chebyshev polynomials
Here is how to prove it with more standard methods. First of all, let me
restate your identity:
Definition. Let $\mathbb{N}=\left\{ 0,1,2,\ldots\right\} $. A
partition shall mean an integer
...
- 33.8k
6
votes
Curious identity between the two kinds of Chebyshev polynomials
I think I can sketch a shorter proof.
Let $z_j = x_j+x_j^{-1}$, and let $p_m$ and $h_m$ denote the power-sum and complete homogeneous symmetric polynomial.
Then (see e.g p.3 in this preprint)
$$
2 ...
- 15.8k
5
votes
Tight upper bounds on trigonometric polynomials
This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a ...
- 13k
5
votes
Accepted
Closed form of $\prod_{k=1}^{n}\left(\cos(kx)-1\right)$
$$\prod_{k=1}^{n}\left(\cos kx-1\right)=2^{-n} e^{-\frac{1}{2} i n (n+1) x} \left(e^{i x};e^{i x}\right)_n^2,$$
with $(\cdot;\cdot)_n$ the q-Pochhammer symbol.
I guess this counts as a "closed ...
- 188k
5
votes
Rigorous estimates on roots of function
If my computations are correct, there is a root of the form $x=1+\sin^2(\frac\theta2)$ with:
$$\theta \in \left(\frac{k\pi}N,\frac{k\pi}N+\frac\pi{2N}\right) \text{ if }k < \frac{N}3-\frac12$$
$$\...
- 3,808
5
votes
Accepted
Optimization problem on trigonometric polynomials
$f(x)=\frac 12(1+\sin nx)$ is, indeed, the optimal choice. To see it, let's normalize a bit differently by $0\le f\le 2$. Then $f=1+g$ where $g$ is a real trigonometric polynomial of degree $n$ ...
- 61.9k
5
votes
Closed formula for sine powers
Letting $\xi=e^{\frac{2\pi i}{2n}}$, a primitive root of unity $z^{2n}=1$. Then, $\sin\left(\frac{\pi k}n\right)=\frac{\xi^{k}-\xi^{-k}}{2i}$ and hence
\begin{align}
\sum_{k=0}^{n-1}\sin^m\left(\frac{\...
- 43.2k
5
votes
Accepted
Is there a law of cosine for n-dimensional hyperbolic simplex
Yes, something like that is proved in the paper by Simon Kokkendorff:
Kokkendorff, Simon L., Polar duality and the generalized law of sines, J. Geom. 86, No. 1-2, 140-149 (2006). ZBL1115.51010.
It ...
- 96.4k
5
votes
Better trigonometrical inequalities for $\zeta(s)$?
this is an answer to the question as originally posed, without the additional conditions on $a_0$ and $b_0$
for example,
$$6+\cos \theta+2 \cos 2 \theta+3 \cos 3 \theta\geq 0,$$
or more ...
- 188k
4
votes
Accepted
$L_p$ norms of $0-1$ exponential sums
For even integer exponents, say $p=2k$ and $p \geq2$, the quantity is just the $k$-order additive energy of the set $S \subset \mathbb{Z}$ of non-zero Fourier coefficients. It is easy to see that this ...
- 13k
4
votes
Accepted
Fourier coefficients of Selberg polynomials
As it is odd, we can write the Vaaler polynomial $V_K(x)=\sum_{1 \le k \le K}c_{k,K} \sin 2\pi kx$ and its fundamental property is that $|c_{k,K}| \le \frac{1}{\pi k}$.
This follows easily from its ...
- 2,025
3
votes
Rigorous estimates on roots of function
Let
$$a_i=\sin ^2\left(\frac{\pi i}{N}\right)\qquad \text{and} \qquad b_i=1+\sin ^2\left(\frac{\pi i}{2 N}\right)$$ and consider
$$f(x)=1-\frac{1}{N} \sum_{i=1}^N \frac{a_i}{b_i-x}$$ For the root ...
- 1,440
3
votes
Accepted
Orthogonal vectors translation using standard vectors
$\newcommand\v{\mathbf v}\newcommand\w{\mathbf w}$Let us show that the vectors $\w_1,\dots,\w_n$ are linearly independent. Of course, then the first $m$ of these $n$ vectors are linearly independent.
...
- 128k
3
votes
Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?
Let $r$ be an irrational real number. For real $x>0$, let $U_x$ be a random variable (r.v.) uniformly distributed on the interval $[0,x]$, and then let
$$C_x:=\cos rU_x+2\cos U_x.$$ Then the ...
- 128k
2
votes
cosine of rational multiples of Pi take values of equal difference
Let $\zeta=e^{2\pi i/N}$ and $\alpha=e^{2\pi i/(2N)}$. Your assumption was $\Re(\alpha(\zeta^{m_3}-2\zeta^{m_2}+\zeta^{m_1}))=0$. This is equivalent to $\alpha(\zeta^{m_3}-2\zeta^{m_2}+\zeta^{m_1}))+\...
- 23.2k
1
vote
Accepted
A sine type Chebyshev system
$\newcommand\be\beta$The answer is no.
For instance, suppose that $n=2$, $\be_{11}=\be_{12}=\be_{21}=1=-\be_{22}$. Then for
$$D(t_1,t_2):=\det\big(\beta_{kl}\sin(lt_k)\big)_{k,l=1}^n$$
we have $D(1,2)=...
- 128k
1
vote
Polynomial satisfied by $\cos^n(t)$ and $\sin^n(t)$
Here are some small explicit solutions and general comments.
Since the polynomials are symmetric, one would expect them to be more nicely expressed using $$S=x+y \\ M=x-y\\ P=xy.$$
Here $M$ should ...
- 30.1k
1
vote
Reference request: Gaussian almost periodic functions
Estimation for almost periodic processes (2006), provides a general method to determine the lines of support of the spectra, with applications to bias and covariance estimation.
- 188k
1
vote
Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation
This is very broad, and largely depend on some details you have not provided. Usually the questions are -
How smooth is $f$?
Is $P$ a measure with any special properties, e.g., has an associated ...
- 3,574
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