Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{R}^{2m}$ if it solves $\Delta^m \phi=0$. The case $m=1$ is the "classical" harmonic case: it contains the Weistrass theorem (the one dealing with the uniform approximation of $2\pi$-periodic continuous functions by trigonometric polynomials) and it can be seen as a corollary of Runge's theorem in complex analysis.
More specifically, a more modest version of my question could be:
if $\Omega$ is a bounded domain in $\mathbb{R}^{2m}$ with smooth boundary and $\psi$ is some given polyharmonic function in $\Omega$ (say continuous in $\bar{\Omega}$), letting $Z_\psi$ be its zero set in $\bar{\Omega}$, is-it true that the set of (the restrictions of) the polyharmonic functions in $\mathbb{R}^{2m}$ (or in $\bar{\Omega}$ in some sense) are dense in $C^0(Z_\psi)$ for the uniform norm?
Unless I am wrong, using the maximum principle, the case $m=1$ of this more specific question is a corollary of the aforementioned classical Walsh-Lebesgue theorem.
Many thanks in advance for any clue about this!
NB: I found some references for analogs of Runge's theorem in higer dimensions (dealing with Cliffordian functions, but, unless I missed it, it does not clearly solve the present question).
