# Questions tagged [trigonometric-sums]

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16
questions with no upvoted or accepted answers

**10**

votes

**1**answer

1k 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

**0**answers

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 ...

**5**

votes

**0**answers

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}...

**3**

votes

**0**answers

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 $\int_{-\pi}^\pi F_N(t) dt$ ...

**3**

votes

**0**answers

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^{...

**3**

votes

**0**answers

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}}, \,\, \...

**2**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 $b_0$). ...

**2**

votes

**0**answers

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) \...

**1**

vote

**0**answers

122 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 ...

**1**

vote

**0**answers

108 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)}}} \...

**1**

vote

**0**answers

128 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

**0**answers

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\...

**0**

votes

**0**answers

111 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, ...

**0**

votes

**0**answers

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. ...

**0**

votes

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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 ...