  We study approximation properties of weighted $L^2$-orthogonal projectors
onto the space of polynomials of degree less than or equal to $N$ on the unit
disk where the weight is of the generalized Gegenbauer form $x \mapsto
(1-|x|^2)^\alpha$. The approximation properties are measured in Sobolev-type
norms involving canonical weak derivatives, all measured in the same weighted
$L^2$ norm. Our basic tool consists in the analysis of orthogonal expansions
with respect to Zernike polynomials. The sharpness of the main result is proved
in some cases and otherwise strongly hinted at by reported numerical tests.
  A number of auxiliary results of independent interest are obtained including
some properties of the uniformly and non-uniformly weighted Sobolev spaces
involved, a Markov-type inequality, connection coefficients between Zernike
polynomials and relations between the Fourier-Zernike expansions of a function
and its derivatives.
