We study the multiplier algebras obtained as the closure $A(H)$ of the polynomials acting as multipliers on on certain reproducing kernel Hilbert spaces $H$ on the ball of $\mathbb C^d$. We first obtain a complete description of the dual and second dual spaces in terms of the complementary bands of Henkin and totally singular measures. This is applied to obtain several definitive results in interpolation. If $E$ is a compact totally null set, then every $h\in M_n(C(E))$ may be interpolated by a function $f\in M_n(A(H))$ of the same norm. If in addition, $F$ is a finite subset of the open ball, one may simultaneously interpolate values on $F$ and $E$ with optimal supremum norm and near optimal multiplier norm. Moreover if the restriction map from $A(H)$ to $C(E)$ is surjective, then $E$ is totally null. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball.
(This is joint work with Michael Hartz.)