Skip to main content

All Questions

Filter by
Sorted by
Tagged with
1 vote
0 answers
690 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 \...
Max Lonysa Muller's user avatar
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}}, \,\, \...
sokho's user avatar
  • 197
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(...
Brendan McKay's user avatar