Lebesgue constants of the Dirichlet kernel

Remarks

For the estimate, use $\abs{\sin(x/2)} \le \abs{x}/2$ and compare $\abs{\sin((N + \frac{1}{2}) x)}$ on each interval between consecutive zeros with its mean value $\frac{2}{\pi}$. Summing over about $N$ intervals gives a harmonic sum, hence the $\log N$. The norm of $f \mapsto S_N f(0) = \frac{1}{2\pi} \int f(-t) D_N(t) \, dt$ is at most $L_N$, and it equals $L_N$ by testing on continuous functions that approximate $\operatorname{sgn} D_N$. Example: $L_0 = 1$ and $L_N > 1$ for $N \ge 1$; the values grow very slowly, like $0.405 \log N$.

Going deeper

For a student

By the uniform boundedness principle, the unboundedness of $L_N$ implies that some continuous function has a Fourier series that diverges at $0$. It also implies that $S_N f$ does not converge to $f$ in $L^1(\mathbb{T})$ for every $f \in L^1(\mathbb{T})$, because the norm of $S_N$ on $L^1$ is also $L_N$. On the positive side, $\abs{S_N f - f} \le (1 + L_N) E_N(f)$, where $E_N(f)$ is the distance from $f$ to trigonometric polynomials of degree at most $N$; combined with Jackson's theorem this gives the Dini–Lipschitz criterion. The Fejér kernel, which has $L^1$ norm $1$, avoids the problem and gives Fejér's theorem.