= Dirichlet kernel harmonic lower bound
{c}
{title2=$\Lambda_N\geq\frac4{\pi^2}\sum_{j=1}^N1/j$}
For $D_N(t)=\sin((N+1/2)t)/\sin(t/2)$, define $\Lambda_N=(2\pi)^{-1}\int_{-\pi}^{\pi}|D_N|$. Evenness and $\sin(t/2)\leq t/2$ on $(0,\pi]$ give $\Lambda_N\geq(2/\pi)\int_0^{(N+1/2)\pi}|\sin v|/v\,dv$. The first $N$ complete sine periods of length $\pi$ have absolute sine integral two, and on the $j$th such interval $1/v\geq1/(j\pi)$. This proves the displayed unbounded lower bound. The real functional $f\mapsto S_N(f,0)$ on continuous functions has norm exactly $\Lambda_N$: the upper bound is its kernel integral, and the continuous functions $D_N/\sqrt{D_N^2+\delta^2}$ approach that bound as $\delta\downarrow0$ by <dominated convergence>.
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