We consider the space, CRp (M), consisting of CR functions which also lie i
n L-p (M) on a quadric submanifold M of C-n of codimension at least one. Fo
r 1 less than or equal to p less than or equal to infinity, we prove that e
ach element in CRp (M) extends uniquely to an H-p function on the interior
of the convex hull of M. As part of the proof, we establish a semi-global v
ersion of the CR approximation theorem of Baouendi and Treves for submanifo
lds which are graphs and whose graphing functions have polynomial growth.