Questions tagged [trigonometric-sums]
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25 questions with no upvoted or accepted answers
11
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A question on trig series
Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
- 903
7
votes
0
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317
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Multiple Fourier series
In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis on Euclidean Spaces" one states in page 268 the following theorem:
Theorem 1: The trigonometric series
$$\...
6
votes
0
answers
80
views
Delaying the first zero of a trigonometric series
Let $a_1,\ldots,a_n \in [0,1], \omega_1,\ldots,\omega_n \in (0,+\infty)$, and define, for every $t \ge 0$,
$$
f(t) := \sum_{i = 1}^n a_i \sin(\omega_i t).
$$
I'm interested in trying to optimize the ...
- 562
5
votes
0
answers
403
views
Generalization of Pompeiu's theorem
Let us recall the statement of
Pompeiu's theorem. Let $A_1A_2A_3$ be a regular triangle inscribed in a circle $\omega$. Let $X$ be an arbitrary point on the arc $A_1A_3$. Then $$|XA_1|-|XA_2|+|XA_3|=...
- 691
5
votes
0
answers
117
views
Concentration of weighted random chirp
I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have
\begin{align*}
\mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}...
- 859
4
votes
0
answers
150
views
Trigonometric sum and residues
I am interested in the sum
$$
\sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g}
$$
where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as
$$
-1-...
- 195
4
votes
0
answers
289
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
3
votes
0
answers
106
views
A new arranging of discrete sine transform
Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix
$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$
Let us denote by $s_{-,l}$ the $l^{\text{...
- 4,058
3
votes
0
answers
377
views
For which values of $ x$ we have $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges?
The copy of this question is posted here
I have tried to to determine values of real $x$ for which $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges but I can't , Presumably the set of values $\{ ...
3
votes
0
answers
152
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
3
votes
0
answers
119
views
Does the following inequality hold under Zygmund condition?
Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, \...
- 197
2
votes
0
answers
120
views
Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $
Background
The Norwegian mathematician and astronomer Carl Størmer did important work on the equation
$$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$
...
- 4,829
2
votes
0
answers
91
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(Dis)continuity of periodic functions with non-summable Fourier series
Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$.
We assume moreover that the square-summable Fourier coefficients of $f$, ...
- 2,306
2
votes
0
answers
511
views
Conjecture about $\prod_{n=1}^{\infty}\left|1+\frac{\tan\left(n\right)+1}{n\tan\left(n+1\right)+n}\right|=0$
Ps: I have reached the number of question I can propose so let this here please .And it's a work on open questions .
Does we have in radians :
$$\prod_{n=1}^{\infty}\left|1+\frac{\tan\left(n\right)+1}{...
2
votes
0
answers
233
views
Finite sum involving root of unity
I have the following sum:
$$\sum_{\sigma=1,\text{odd}}^{\frac{p}{2}-2}\frac{\sin \frac{r \sigma \pi}{p}}{\sin \frac{q \sigma \pi}{p}}$$
where $p\equiv2\pmod4$, $p$ and $q$ are coprime numbers such ...
- 21
2
votes
0
answers
112
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
2
votes
0
answers
182
views
Why hexagons? The maximal minimum of a sum of cosines on the plane with frequencies on the unit circle
We wish to maximize the minimum of a weighted sum of cosines in the plane, when the frequency components are on the unit circle. Formally:
$$\max_{\{ a_i,\theta_i,\phi_i \}_{i=1}^{N} } \min_{(x,y) \...
- 968
1
vote
0
answers
692
views
What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?
Cross-post from MSE.
The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...
- 4,829
1
vote
0
answers
156
views
Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
- 105
1
vote
0
answers
130
views
How to evaluate this sum of roots of unity with condition to zero
In evaluating the sum:
$$\tag{1}\label{e1}\sum\limits_{j = 1 < h}^N {\left( {{e^{\frac{{i\pi }}{N}\left( {{s_1}j + {s_2}h} \right)}} - {e^{\frac{{i\pi }}{N}\left( {{s_1}h + {s_2}j} \right)}}} \...
- 59
1
vote
0
answers
168
views
How to prove that the convergence of $\sum_{n=1}^{\infty} \frac{\sec^a n}{n^c}$ implies that of $\sum_{n=1}^{\infty} \frac{\csc^a n}{n^c}$
The most general thing I've gotten is that the absolute convergence of $$\sum_{n=1}^{\infty} \frac{\csc^a (n + x)}{n^c}$$ implies that of $$\sum_{n=1}^{\infty} \frac{\csc^a \left(\frac{m}{2} n + \...
- 602
1
vote
0
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38
views
Boundedness of partial products for a divergent trig product
I am looking at a discrete dynamical system and I wish to show that it is bounded. I know that the displacement after $n$ iterations is given by the product
$$\Delta_n=\prod_{k=0}^n \left(1+\frac{2\...
- 11
0
votes
0
answers
125
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$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \sin(\mu E)\cos(\mu F) \geq 0$ - Version 2.0
I recently asked this question here Inequality involving sine and cosine
It turned out that with the conditions I required there, the inequality does not hold. I tried to add extra conditions now, ...
- 447
0
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0
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Can we solve for $c$ in the equation $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0$?
This question was posted on stackexchange, but with no response. So, I thought it appropriate to post it here too. Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. ...
- 101
0
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0
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Linearizing a multifrequency signal
I have a component of a signal
$$\sin (k\omega_1t + \ell\omega_2t)$$
with wavenumbers $k, \ell \in \mathbb{Z}$, frequencies $\omega_1, \omega_2 \in \mathbb{R^+}$ and time $t \in \mathbb{R^+}$. (This ...
