# Questions tagged [trigonometric-sums]

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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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\int_{-\pi}^\pi F_N(t) dt$ ...
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### 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} ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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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: $$\max_{\{ a_i,\theta_i,\phi_i \}_{i=1}^{N} } \min_{(x,y) \... 2 votes 1 answer 378 views ### Maximal minimum for a sum of two (or more) cosines Please prove (or disprove, and give the correct answer):$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right] $$In other words, find r \geq 1, ... 4 votes 2 answers 553 views ### Non-standard Gauss sums I have the following problem. Let p be some prime. What is the value of \begin{equation} \sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl}, \end{equation} where \left(\frac{k+1}{p}\right) ... 5 votes 2 answers 261 views ### About trigonometric series of the Lip \alpha class Assume we have a trigonometric series$$ 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} |...
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 ...