If D_m is the mth Dirichlet kernel, $||D_m||_1\to\infty$ as $m\to\infty$

Question

I was working on this problem for my own studying but am stuck on how to solve it. Let $D_m$ be the $m$th Dirichlet kernel. Show that $||D_m||_1\to\infty$ as $m\to\infty$. Anything would help. Thanks

Answer 1

\begin{align*} \int_{-\pi}^{\pi}|D_m(t)|\,dt&=2\int_{0}^{2\pi}\left|\frac{\sin((m+1/2)t)}{\sin(t/2)}\right|\,dt\\ &\geq2\int_{0}^{2\pi}\frac{|\sin((m+1/2)t)|}{t/2}\,dt\\ &=4\int_0^{(m+1/2)\pi}\frac{|\sin s|}{s}\,ds\\ &\geq4\sum_{j=1}^m\int_{(j-1)\pi}^{j\pi}\frac{|\sin s|}{s}\,ds\\ &\geq4\sum_{j=1}^m\int_{(j-1)\pi}^{j\pi}\frac{|\sin s|}{j\pi}\,ds\\ &=\frac{4}{\pi}\sum_{j=1}^m\frac{1}{j}\int_0^{\pi}\sin s\,ds\\ &=\frac{8}{\pi}\sum_{j=1}^m\frac{1}{j}\\ &\to\infty\quad(m\to\infty) \end{align*} By the way, it tends slowly to $\infty$ (indeed, one can show that it does so as $\log m$).