Let $f(\alpha) = |\{\alpha\}-1/2|$. What is the trigonometric polynomial $F_N$ of degree $N$ (i.e., a linear combination $\sum_{n=-N}^N a_n e(\alpha n)$, $a_n\in \mathbb{C}$, where $e(r)= e^{2\pi r}$) such that

(i) $F_N(\alpha)\geq f(\alpha)$ for all $\alpha\in [0,1]$,

(ii) $\int_0^1 F_N(\alpha) d\alpha$ is minimal?

Note: there's a result by Vaaler close to this (for $\{\alpha\}-1/2$ instead of $|\{\alpha\}-1/2|$).

Let $f(\alpha) = |\{\alpha\}-1/2|$. What is the trigonometric polynomial $F_N$ of degree $N$ (i.e., a linear combination $\sum_{n=-N}^N a_n e(\alpha n)$, $a_n\in \mathbb{C}$, where $e(r)= e^{2\pi i r}$) such that

(i) $F_N(\alpha)\geq f(\alpha)$ for all $\alpha\in [0,1]$,

(ii) $\int_0^1 F_N(\alpha) d\alpha$ is minimal?

Note: there's a result by Vaaler close to this (for $\{\alpha\}-1/2$ instead of $|\{\alpha\}-1/2|$).