Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$ $$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\sum_{k\ge1}a_k\cdot k\cos(kx)\right) = \left(\sum_{k\ge1}a_k\sin(kx)\right)\left(\sum_{k\ge1}a_k\cos(kx)\right).$$ What can we say about $\{a_k\}$? Can we show that there exists at most one term of $a_k$ that is nonzero? Note that if we set $w(x) = -\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)$, we have the relation $u_xw = uw_x$ and $w_x = Hu$. Here $H$ denotes the standard Hilbert transform.

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