Maximum value of a mapping on a compact subset of $\ell^{2}$

Question

Let $A = \{x \in \ell^{2}: \sum_{n = 1}^{\infty}n|x_{n}|^{2} \leq 1\}$. What is the largest value $\frac{1}{2\pi}\int_{0}^{2\pi}\left|\sum_{n = 1}^{\infty}x_{n}e^{in\theta}\right|\, d\theta$ can take on $A$?

Answer 1

Define $f:[0,2\pi]\to\mathbb{C}$ as: $$f(\theta)=\sum_{n=1}^{+\infty}x_n e^{ni\theta}.$$ Then $f$ is a $L_2$-function with zero mean over $(0,2\pi)$, and we have: $$\|f\|_1 \leq \sqrt{2\pi}\,\|f\|_2 \tag{1}$$ due to the Cauchy-Schwarz inequality and $$\|f\|_2 \leq \|f'\|_2\tag{2}$$ due to Wirtinger's inequality, or just Parseval's identity. $(1)$ and $(2)$ give: $$\frac{1}{2\pi}\int_{0}^{2\pi}\left|\sum_{n=1}^{+\infty}x_n e^{ni\theta}\right|\,d\theta\leq\frac{1}{\sqrt{2\pi}}.\tag{3}$$ Probably this can be improved a little, since equality cannot hold both in $(1)$ and in $(2)$.