Questions tagged [trigonometric-sums]

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-4
votes
0answers
19 views

Could someone explain the range extension part? [closed]

enter image description here strong text Why is it out of range and why is it minus pi instead of adding pi as it would also give the same value of tany **
8
votes
0answers
145 views

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(\...
5
votes
2answers
369 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 $$\...
1
vote
2answers
348 views

Order of magnitude for trigonometric sum

Consider the following sum: $$ S_N = \sum_{ \substack{ k_1 + k_2 + k_3 =N \\ -(N-2) \leq k_1, k_2 , k_3 \leq N \\ k_1, k_2 , k_3 \neq 0 } } \frac{1}{k_1 k_2 k_3} \sin\left( \frac{k_1\pi}{N} \right) \...
0
votes
0answers
106 views

$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \sin(\mu E)\cos(\mu F) \geq 0$ - 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, ...
3
votes
1answer
565 views

Inequality involving sine and cosine

I am trying to prove that given $A,B,C,D,E,F \in \big]0,\frac{\pi}{2}\big]$ fixed and $A+C \geq E$ and the following equation holds for $\mu = 1$: $$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \...
29
votes
3answers
2k views

A conjectural trigonometric identity

Recently, I formulated the following conjecture which seems novel. Conjecture. For any positive odd integer $n$, we have the identity $$\sum_{j,k=0}^{n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac{n^2}...
3
votes
1answer
120 views

Trigonometric cancellation on the unit circle

Let $z \in \mathbb{C}$ with $|z|=1$ and $z\ne 1$. Now consider the sum $$S(N,p)=\sum_{k=0}^N k^p z^k,$$ for some positive integers $N,p$. An immediate upper bound on $|S(N,p)|$ is $$|S(N,p)|\le C_1(...
2
votes
0answers
185 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 ...
6
votes
3answers
374 views

The first zero-crossing of a combination of sines

Let $\{c_i\}_{i=1}^n$ be a sequence of real numbers such that $c_i \geq 0$ for each $i$ and $\sum_{i=1}^n c_i = 1$. Let $\omega_i \in [\delta, \Delta]$ for each $i$, where $\delta$ and $\Delta$ are ...
6
votes
1answer
276 views

Asymptotic behavior of a certain trigonometric partial sum

Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum: $$ f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...
1
vote
1answer
105 views

Proof of an inequality $s_m(n) \le f_m(n)$

For fixed $m = 0, 1, 2, ...$ $$f_m(k) = \prod_{j=1}^{m}(k+j).$$ Some examples of $f_m(k)$ are as following: $$f_0(k) = 1, \quad f_1(k) = (k+1), \quad f_2(k) = (k+1)(k+2).$$ The $s_m(n)$ is defined as ...
3
votes
2answers
211 views

Inductive proof of $s(n)≤n+1$

I was able to conclude, numerically, the following: $$s(n) = \sum_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum_{k=j}^n\frac{(k+1)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2}\le n+1$$ for $x\in[0,1]$. For example ...
1
vote
0answers
116 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 ...
3
votes
0answers
202 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 $\int_{-\pi}^\pi F_N(t) dt$ ...
0
votes
1answer
181 views

Finding closed form expression for the roots of $f(x) = \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}$

Let us define function $f:[0~ 2\pi] \rightarrow R$ as follows: \begin{align} f(x)\triangleq \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}, \end{align} ...
2
votes
0answers
89 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 $b_0$). ...
4
votes
1answer
230 views

Identities for Chebyshev polynomials of the second kind

While calculating an integral in a quantum mechanical problem by two different methods, I came across the following identity $$\sum_{k=0}^n\sum_{m=0}^{2k}(-2)^m\binom{2(n-k)}{n-k}\binom{2k}{k}\binom{...
5
votes
3answers
450 views

positive sum of sines

This was asked but never answered at MSE. Let $f(x) = \sin(a_1x) + \sin(a_2x) + \cdots + \sin(a_nx)$, where the $a_i$'s represent distinct positive integers. Suppose also that $f(x)$ satisfies the ...
1
vote
0answers
107 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)}}} \...
5
votes
0answers
71 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 ...
6
votes
2answers
2k views

Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$

Let $\alpha$ and $\beta$ be incommensurate real numbers. Consider the function $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$ and its positive zeros $x_k(\alpha,\beta)$. Fix $\alpha$ and ...
7
votes
2answers
798 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 ...
6
votes
3answers
533 views

Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins

(Update): Courtesy of Myerson's and Elkies' answers, we find a second simple cyclic quintic for $\cos\frac{\pi}{p}$ with $p=10m+1$ as, $$F(z)=z^5 - 10 p z^3 + 20 n^2 p z^2 - 5 p (3 n^4 - 25 n^2 - 625) ...
0
votes
0answers
64 views

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. ...
1
vote
0answers
127 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 + \...
1
vote
1answer
282 views

Is this function $\mathbb{Q}$ periodic?

Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ? $$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; \sum\...
5
votes
1answer
296 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_\...
12
votes
1answer
580 views

Summation of series involving $\sinh$ of a square root

Consider the following series: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical ...
1
vote
0answers
36 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\...
2
votes
0answers
162 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) \...
2
votes
1answer
276 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 $r \geq 1$, ...
4
votes
2answers
471 views

Non-standard Gauss sums

I have the following problem. Let $p$ be some prime. What is the value of \begin{equation} \sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl}, \end{equation} where $\left(\frac{k+1}{p}\right)$ ...
5
votes
2answers
212 views

About trigonometric series of the Lip $\alpha$ class

Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} |...
0
votes
0answers
34 views

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 ...
2
votes
1answer
196 views

trigonometric sum and inequalities

let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq N}}{\...
4
votes
1answer
204 views

Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...
3
votes
0answers
131 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^{...
5
votes
0answers
110 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}...
6
votes
2answers
738 views

Geometric explanation of Hutton's formula?

$$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$ Is there some geometric construction that explains this beautiful equation (known as Hutton's formula)? Perhaps a "proof without ...
3
votes
0answers
112 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}}, \,\, \...
3
votes
1answer
543 views

Convergence of a Trigonometric Series

After working with a Fourier series for a while, I realized that it would be of great help to me if I could prove that the following limit is zero: $$\lim_{N\to\infty}\frac{1}{N}\sum_{m=1}^\infty\frac{...
6
votes
2answers
834 views

Estimating a sum of gauss sums

Hey guys, I'm concerned with bounding the following sum of gauss sums from above $$\sum_{p\leq x}~{\frac{1}{(p-1)^2}}\sum_{m=1}^{p-1}~\sum_{\chi~(p)}~\sum_{a=1}^{p-1}{~\chi^m(a)e\left(\frac{a}{p}\...
9
votes
2answers
764 views

Need to bound a trigonometric sum

Let $\boldsymbol{\theta}=(\theta_1,\ldots,\theta_m)$ be a vector of real numbers in $[-\pi,\pi]$. For $t\ge 0$, define $$ f(t,\boldsymbol{\theta}) = \binom{m+t-1}{t}^{-1} \sum_{j_1+\cdots+j_m=t} \exp(...
7
votes
2answers
737 views

Salie-type sum bound

I am interested in bounding the following Salie-type ("twisted Kloosterman") sum $$ S(a,b,\beta) = \sum_{x \in \mathbb{Z}/{p^{\beta}}\mathbb{Z}} \left( \frac{x}{p^{\beta}} \right) \chi(ax + bx^{-1}). ...
3
votes
1answer
414 views

Summations in $\tan^2$

Hey all, I was just wondering if anyone had come across the following identities, valid for $m\in\mathbb{N}$. I've used Abramowitz and Stegun, Maple, Mathematica etc but can't find them anywhere. I ...
4
votes
1answer
1k views

Trigonometric identities

In the rant I wrote at http://ncatlab.org/nlab/show/trigonometric+identities+and+the+irrationality+of+pi I asked: Are these four identities the first four terms in a sequence that continues? This ...