Polynomial interpolation and residue currents

We show that a global holomorphic section of $\mathcal{O}(d)$ 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 ${\mathcal{O}}_X$. 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 $d$.