How does a fourier invariant representation of spherical harmonics look like?

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The Hermite functions https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions are fourier invariant up to scaling. How does a fourier transform invariant representation of spherical harmonics look like? The Laplace's spherical harmonics ${\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}$ expanded in Hermite polynomials?