Can we approximate $f(x)=-x_1^2/2$ defined in the unit cube of $\mathbb{R}^2$ by a sequence of harmonic polynomials?
We can find a compact subset $K$ of the unit cube such that it has empty interior, the complement of it in the unit cube has measure as small as you want and the complement of it in $\mathbb{R}^2$ is connected, so by Mergelyan's theorem, we can find a sequence of polynomials approximating $f$ uniformly in $K$. But how we find a sequence of harmonic polynomials approximating $f$ uniformly? Thanks for any help!