  Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$
\CW_{\a,\b,\g}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} (1-x^2)^\g(1-y^2)^{\g}, $$
are studied and shown to be related to the Koornwinder polynomials defined on
the region bounded by two lines and a parabola. In the case of $\g = \pm 1/2$,
an explicit basis of orthogonal polynomials is given in terms of Jacobi
polynomials and a closed formula for the reproducing kernel is obtained. The
latter is used to study the convergence of orthogonal expansions for these
weight functions.
