The function $f(x)$ that $\mathcal{F}\left[f(x)^3\right]\propto\left[\tilde{f}(k)\right]^3$

Question

I have been trying to find an even square integrable real function $f(x)$ that $\int_{-\infty}^{\infty}f(x)^2dx=1$ and the fourier transform of its third power is proportional to the third power of its fourier transform: $$\mathcal{F}\left[f(x)^3\right]\propto\left[\tilde{f}(k)\right]^3$$ Does this kind of function exist? How to find it?

Answer 1

Every $f\in L^2(\mathbb{R})$ can be approximated by a series of Hermite functions. So suppose $f(x)=p_n(x)e^{-x^2/2}$ is such an approximation. Then $$\hat{f}(\xi)=q_n(\xi)e^{-\xi^2/2}$$ where $q_n$ is some other polynomial of degree $n$.

If $\hat{f}^k=c\widehat{f^k}$ were to hold then $$q_{kn}(\xi)e^{-k\xi^2/2}=c\tilde{q}_{kn}(\xi)e^{-\xi^2/2k}$$

Looking at the behaviour as $\xi\to\infty$ shows that non-trivial solutions are only possible if $k=1$.