# Questions tagged [trigonometric-sums]

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### 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 ...
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### 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 $b_0$). ...
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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, ...
### 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. ...
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 ...