A friend of mine raise an interesting question.
Let $V$ be a dense subset in $L^2(\Bbb R)$. The choice of $V$ is completely under our control, we can even take $V=\cal D(\Bbb R)$ or $V=\cal S(\Bbb R)$.
Let also $T$ be a linear operator defined on $V$ such that it acts locally on both the physical space on the Fourier space.
I wonder if we can prove that $$T = \sum_{k=1}^{n}p_k(x)\partial_x^k$$ where $p_k(x)$ is a polynomial. If we can, was it shown already? If we can not, what are the counterexamples or the reasoning behind this negative result?
I'll be glad to hear all suggestions.