All Questions
Tagged with trigonometric-sums fourier-analysis
10 questions
9
votes
2
answers
799
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
$$ f(t,\boldsymbol{\theta}) = \binom{m+t-1}{t}^{-1}
\sum_{j_1+\cdots+j_m=t} \exp(...
- 37.7k
7
votes
0
answers
317
views
Multiple Fourier series
In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis on Euclidean Spaces" one states in page 268 the following theorem:
Theorem 1: The trigonometric series
$$\...
6
votes
3
answers
441
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 ...
- 61
5
votes
1
answer
244
views
Smallest regular $m$-gon covering a regular $n$-gon
I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.
Let us fix a regular $n$-gon with area $1$. What is the smallest ...
- 1,889
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
3
votes
1
answer
147
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
$$|S(N,p)|\le C_1(...
- 265
3
votes
0
answers
119
views
Does the following inequality hold under Zygmund condition?
Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, \...
- 197
2
votes
0
answers
91
views
(Dis)continuity of periodic functions with non-summable Fourier series
Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$.
We assume moreover that the square-summable Fourier coefficients of $f$, ...
- 2,306
1
vote
1
answer
319
views
Is this function $\mathbb{Q}$ periodic?
Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ?
$$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; \sum\...
- 1,199
1
vote
0
answers
692
views
What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?
Cross-post from MSE.
The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...
- 4,829