Show that there is $C_1,C_2$ s.t. $C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$ where $D_N$ is the dirichelet kernel.

Question

Let $D_N=\frac{\sin(\pi(2N+1)x)}{\sin(\pi x)}$ the dirichlet kernel. Show that there is $C_1,C_2$ s.t. $$C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$$ where $\mathbb S^1=\mathbb R/\mathbb Z$ and $$\|D_N\|_{L^1(\mathbb S^1)}=\int_{\mathbb S^1} |D_N(x)|dx.$$

I have no idea on how to proceed. Any idea would be welcome.

Answer 1

Upper bound

The contribution of $|x|<1/N$ is $O(1)$. The contribution of $|x|>1/N$ is bounded by $\int_{1/N}^\pi x^{-1}\,dx = O(\log N)$.

Lower bound

Consider the set where $|\sin(\pi(2N+1)x)|\ge 1/\sqrt{2}$. It's a union of some intervals $[a_n, b_n]$. Note that $b_n-a_n = a_{n+1}-b_n$. Also, the denominator on $[a_n, b_n]$ is about constant, $\sim 1/b_n$. And the same can be said about $[b_n,a_{n+1}]$. The conclusion is that integrating $1/|\sin \pi x|$ over $\bigcup_n [a_n,b_n]$ yields $\ge c\int_{a_1}^{\pi}x^{-1}\,dx$ with some $c>0$. This evaluates to $\Omega(\log(1/a_1))=\Omega(\log N)$.