  We show that a global holomorphic section of $\mathscr{O}(d)$ restricted to a
closed complex subspace $X \subset \mathbb{P}^n$ has an interpolant if and only
if it satisfies a set of moment conditions that involves a residue current
associated with a locally free resolution of $\mathscr{O}_X$. When $X$ is a
finite set of points in $\mathbb{C}^n \subset \mathbb{P}^n$ this can be
interpreted as a set of linear conditions that a function on $X$ has to satisfy
in order to have a polynomial interpolant of degree at most $d$.
