Questions tagged [trigonometric-polynomials]
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55
questions
4
votes
1
answer
153
views
Fourier coefficients of Selberg polynomials
In Montgomery's "Ten Lectures on the Interface Between Number Theory and Harmonic Analysis" a bound for the Fourier coefficients of the Selberg polynomial $S^+_K$ is obtained by using what ...
- 3,497
1
vote
0
answers
76
views
Showing Vaaler polynomial is a good approximation to saw tooth function
Vaaler's polynomial is defined
$$
V_K(x) = \frac{1}{K+1}\sum_{k=1}^K\left(\frac{k}{K+1} - \frac12\right) \Delta_{K+1}\left(x - \frac{k}{K+1}\right) +
\frac{1}{2 \pi (K+1)}\sin 2 \pi (K+1) x - \frac{1}{...
- 3,497
1
vote
0
answers
50
views
Sum of absolute values of trigonometric polynomials
I am trying to tackle the following problem:
Let $A_{f},A_{g} \in \mathbb{R}^{3 \times 3}$ be symmetric matrices and let $f: [-\pi,\pi)^{2} \to \mathbb{R}$ and $g: [-\pi,\pi)^{2} \to \mathbb{R}$ ...
- 19
0
votes
0
answers
52
views
Prove the existence of a certain number of real solutions to a real coefficient nonlinear algebraic system
The factorization of a real element $3\times 4$ matrix $P=(p_{i,j})$, was finally converted into the problem of solving a $9\times 9$ nonlinear polynomial system as below.
All $p_{i,j}$'s are ...
- 205
0
votes
0
answers
84
views
Closed formula for iterated Fourier series
I'm trying to obtain a closed formula for the following integral.
\begin{align}
I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt]
& {}+ b_{1,...
- 253
7
votes
3
answers
513
views
Rigorous estimates on roots of function
We consider the function
$$f(x) = 1- \frac{1}{N} \sum_{i=1}^N \frac{\sin\left(\tfrac{\pi i}{N}\right)^2}{1+\sin\left(\tfrac{\pi i}{2N}\right)^2-x}.$$
The arguments of the two sines differ by a factor ...
5
votes
0
answers
202
views
Majorizing $|\{\alpha\}-1/2|$ by trigonometric polynomials
Let $f(\alpha) = |\{\alpha\}-1/2|$. What is the trigonometric polynomial $F_N$ of degree $N$ (i.e., a linear combination $\sum_{n=-N}^N a_n e(\alpha n)$, $a_n\in \mathbb{C}$, where $e(r)= e^{2\pi i r}$...
- 18.9k
1
vote
0
answers
83
views
Absolute lower bound on derivative of generalized trigonometric polynomial at zeroes
By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\...
- 709
20
votes
1
answer
1k
views
(update) Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
Problem: Given three positive integers $0 < n_1 < n_2 < n_3$ such that
$$n_1 + n_2 \ne n_3, \quad n_2 \ne 2n_1, \quad n_3 \ne 2n_1, \quad n_3 \ne 2n_2,$$
is there always a real number $x$ ...
- 798
5
votes
1
answer
384
views
Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
Problem: Given three positive integers $0 < n_1 < n_2 < n_3$. Is there always a real number $x$ such that
$$\cos n_1 x + \cos n_2 x + \cos n_3 x < -2?$$
- 798
0
votes
0
answers
132
views
What is the inverse Fourier transform of $\operatorname{sinc} \Big{(} \prod_{k=1}^{m} \big{(} k(z-m+1) -z) \big{)} \Big{)} $?
For a certain interpolation problem, I'm looking into a sequence of functions of the form $$f_{m}(z) = \operatorname{sinc} \Bigg{(} \prod_{k=1}^{m} \big{(} k(z-m+1) -z) \big{)} \Bigg{)} . $$
Here, $m&...
- 3,968
10
votes
1
answer
482
views
Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots
Let $P(z) = \prod_{i = 1}^n (z - z_i) \in \mathbb{C}[z]$ be a monic polynomial having all roots $z_1, \dots, z_n$ on the unit circle $\mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$.
What is known about ...
- 121
1
vote
0
answers
130
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 ...
- 9,663
2
votes
1
answer
300
views
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.
- 365
8
votes
2
answers
382
views
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 ...
user215601
45
votes
1
answer
3k
views
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 ...
- 104k
2
votes
1
answer
215
views
$L_p$ 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 ...
- 2,208
6
votes
0
answers
356
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 ...
- 111
0
votes
0
answers
126
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{\...
1
vote
0
answers
142
views
Solutions of equation $\sin \pi x_1\sin \pi x_2=\sin \pi x_3\sin \pi x_4$ [closed]
I am interested in finding all the solution $(x_1,x_2,x_3,x_4)\in \mathbb{R}^4$ of equations:
$$\sin \pi x_1\sin\pi x_2=\sin \pi x_3\sin\pi x_4.$$
I have found out a paper: Rational products of sines ...
- 1,855
14
votes
2
answers
481
views
Curious identity between the two kinds of Chebyshev polynomials
I have found, by accident, an identity that relates a sum of Chebyshev polynomials of the first kind to a Chebyshev polynomial of the second kind. It goes as follows:
Given an integer partition of $n$...
- 243
8
votes
2
answers
924
views
Better trigonometrical inequalities for $\zeta(s)$?
The inequality $$3 + 4 \cos \theta + \cos 2 \theta \geq 0$$ plays a key role in the proof of the classical zero-free region of the Riemann zeta function. Are there other inequalities of the form
$$\...
- 18.9k
-2
votes
1
answer
100
views
What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]
sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there:
How can $\sqrt{2x^2}$ become $4x^2$ ?
- 99
9
votes
3
answers
827
views
Polynomial satisfied by $\cos^n(t)$ and $\sin^n(t)$
For any even $n$, there should be a polynomial $f(x,y)$ that vanishes on the points $(\cos^n(t),\sin^n(t))$ for all $t$ (since it is the image of the projective variety $x^2+y^2 = 1$ under the $n/2$th-...
- 188
5
votes
1
answer
387
views
Optimization problem on trigonometric polynomials
I would like to maximize
$$
\int_0^{2\pi} \frac{(f'(x))^2}{f(x)}dx
$$
subject to $f(x)\leq 1$ for all $x$
over the space of nonnegative trigonometric polynomials of degree smaller or equal to $n$.
...
- 398
1
vote
1
answer
98
views
Reference request: Gaussian almost periodic functions
Let $X(x),x\in R^d$, be a stationary gaussian process for which the covariance function $E(X(0)X(x))=C(x)$ is "almost periodic".
Almost periodic means roughly that $C$ is uniformly approximable by ...
- 1,268
1
vote
1
answer
222
views
Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation
This is a restated version of my original very broad question.
Let $P$ be probability a measure on an interval $[a,b]$ ($-\infty<a<b<\infty$) that's dominated by Lebesgue measure. Let $\...
4
votes
0
answers
256
views
A uniform Riemann sum approximation of the integral of the Fejer kernels
Let $F_N(t)$ denote the Fejer kernel
$$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$
Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...
- 136
4
votes
0
answers
238
views
Reverse Markov-Bernstein inequality for trigonometric polynomials
Let $r(t)$ be a real trigonometric polynomial of degree $n>1$. Assume it has zero at $t=0$ of multiplicity $k>0$. What can be said about the lower bound of the constant $c(k,n)$ such that
$$
\...
- 3,671
2
votes
0
answers
108
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 $b_0$). ...
- 151
3
votes
0
answers
120
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 ...
- 18.9k
10
votes
1
answer
787
views
An extremal problem related either to an uncertainty principle on the circle, or else to the prime number theorem
Consider for $X = 1,2, \ldots$ the set $\mathcal{S}_X$ of trigonometric polynomials $f(t) := \sum_{|k| \leq X} c_k e^{2\pi i kt}$ on the circle $\mathbb{T} := \mathbb{R}/\mathbb{Z}$ of degree $\leq X$ ...
- 13.6k
1
vote
1
answer
287
views
Is there a law of cosine for n-dimensional hyperbolic simplex
We know that, given an n-dimensional Euclidean simplex, for all $1\leq i,j,k,l\leq n+1$, we have(law of sines)$$\frac{A_i A_j}{A_k A_l}=\frac{c_{ij}}{c_{kl}}$$(from Elementary Formulas for a ...
- 97
2
votes
0
answers
1k
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
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
14
votes
0
answers
566
views
Precise form of the mean motion theorem
Consider an exponential polynomial
$$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\...
- 86.4k
4
votes
2
answers
520
views
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: ...
- 59
7
votes
2
answers
832
views
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 ...
- 9,297
3
votes
0
answers
292
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)...
- 176
7
votes
1
answer
308
views
L1 analog of Bernstein's inequality
Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that
$$
\|q\|_1 \leq O(n) \|p\|_1
$$
where we define $\|f\|_p := \left(\int_{-1}^1 |f(x)|^pdx\...
- 73
6
votes
1
answer
384
views
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_\...
- 363
2
votes
0
answers
101
views
Octahedron and System of trigonometric equations
Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
$$\sum_{k=...
- 309
2
votes
1
answer
517
views
Maximal minimum for a sum of two (or more) cosines
Please prove (or disprove, and give the correct answer):
$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right]
$$
In other words, find $r \geq 1$, ...
- 866
9
votes
1
answer
507
views
$L^1$ norm of exponential sum of $n^2 x$
What is the asymptotic order of
$$
\int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.
- 2,179
4
votes
1
answer
138
views
Estimate self crossings of a curve parameterized by a trigonometric polynomial
Given z on the unit circle, let $P(z)= \sum\limits_{k=-D}^D p_k z^k $.
Can one estimate the number of self crossings of the following curve with an analytic expression in terms of the coefficients $\{...
- 81
10
votes
0
answers
536
views
Reciprocal polynomials with roots off the unit circle
A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...
- 95k
3
votes
0
answers
147
views
Bounding expected value of maximum of dot product with random chirp
Let $\mathbf{x}\in\mathbb{C}^n$ with $\|\mathbf{x}\|=1$ with $n<\frac{N}{2}$. I am interested in a bound of the form
\begin{equation*}
\mathbb{E}\Big\{\max_{k\in\{1,2,\ldots,n\}}\Big|\sum_{a=1}^ne^{...
- 859
5
votes
0
answers
116
views
Concentration of weighted random chirp
I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have
\begin{align*}
\mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}...
- 859
7
votes
1
answer
389
views
A conjecture about the measure estimates of a trigonometric polynomial
Formulation of the Conjecture
Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( a_{kj}\sin(jt)+...
- 151
1
vote
0
answers
103
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 ...
- 949
1
vote
0
answers
289
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&...
- 29