All Questions
Tagged with trigonometric-sums real-analysis
11 questions
21
votes
2
answers
2k
views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...
- 35.5k
1
vote
3
answers
183
views
Evaluating a sinusoidal series
Define the sequence of functions
$$f_n(x)=\sum_{m=n}^\infty(-1)^m\frac{x^{2m}}{(2m+1)!} {m \choose n} $$
Is there a closed form expression for arbitrary $n$? It is clear that the result should assume ...
- 17k
9
votes
2
answers
440
views
How to prove this sum involving powers of cosec is an integer?
It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$.
$F(m,N)=\frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\...
- 201
7
votes
2
answers
340
views
Sum of $\sin$ when angles shrink by $1/n$
There are many identities known like
$$\sum_{k=0}^{n-1} \sin (k \cdot \theta + \varphi) = \frac{\sin\left(n \cdot \frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \cdot \sin \left(\frac{2 \...
- 749
4
votes
1
answer
1k
views
The cotangent sum $\sum_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n$
On the Wolfram Research Reference page for the cotangent function (https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/), I saw the following partial sum formula
$$\sum_{k=0}^{n-1}(-1)^k\cot\...
- 128k
11
votes
0
answers
2k
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 $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
- 441
1
vote
0
answers
156
views
Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
- 105
4
votes
0
answers
289
views
A uniform Riemann sum approximation of the integral of the Fejer kernels
Let $F_N(t)$ denote the Fejer kernel
$$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$
Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...
- 136
5
votes
3
answers
787
views
positive sum of sines
This was asked but never answered at MSE.
Let $f(x) = \sin(a_1x) + \sin(a_2x) + \cdots + \sin(a_nx)$, where the $a_i$'s
represent distinct positive integers. Suppose also that $f(x)$ satisfies the ...
- 4,713
3
votes
0
answers
119
views
Does the following inequality hold under Zygmund condition?
Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, \...
- 19.6k
4
votes
1
answer
216
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 ...
- 34.7k