  Weighted pluripotential theory is a rapidly developing area; and Callaghan
\cite{Callaghan} recently introduced $\theta$-incomplete polynomials in \cd for
$d>1$. In this paper we combine these two theories by defining weighted
$\theta$-incomplete pluripotential theory. We define weighted
$\theta$-incomplete extremal functions and obtain a Siciak-Zahariuta type
equality in terms of $\theta$-incomplete polynomials. Finally we prove that the
extremal functions can be recovered using orthonormal polynomials and we
demonstrate a result on strong asymptotics of Bergman functions in the spirit
of \cite{BermanCn}.
