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Let T be the operator $(2\pi)^ \frac{-n}{2}\mathcal{F}$.

Show that, for $k = 0, 1, 2, 3,$ there exists a polynomial $p_k(x)$ of degree $k$ and a complex number $c_k$ such that $T(p_k(x)e^\frac{-x^2}{2}) = c_kp_k(\xi)e^\frac{\xi^2}{2}.$ What is the value of $c_k$?

Let $b$ be a complex number of norm $1$, but not a fourth root of unity. Show that there is no Schwartz function $g$ such that $Tg = bg$.

I am stuck on this question and don't know how to proceed. Could anyone show me how to do it?

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Hints: Use that $\mathcal{F} (x_i f) = i\partial_{\xi_i} \mathcal{F}(f)$ for the first part, repeating as needed.

For the second part, look at iterations $\mathcal{F}\mathcal{F}... (f)$. Eventually you get back $f$.

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