Let $H^2_m$ be the Hilbert function space on the unit ball in
$\C{m}$ defined by the kernel $k(z,w) = (1-\langle z,w
\rangle)^{-1}$. For any weak zero set of the multiplier algebra of
$H^2_m$, we study a natural extremal function, $E$. We investigate
the properties of $E$ and show, for example, that $E$ tends to $0$ at
almost every boundary point. We also give several explicit examples
of the extremal function and compare the behaviour of $E$ to the
behaviour of $\delta^*$ and $g$, the corresponding extremal function
for $H^\infty$ and the pluricomplex Green function, respectively.