Weighted Sobolev orthogonal polynomials and approximation in the ball

We establish simultaneous approximation properties of weighted first-order Sobolev orthogonal projectors onto spaces of polynomials of bounded total degree in the Euclidean unit ball. The simultaneity is in the sense that we provide bounds for the projection error in both a weighted $\mathrm{L}^2$ norm and a weighted $\mathrm{H}^1$ seminorm, both involving the same weight of the generalized Gegenbauer type $x \mapsto (1-\lVert x \rVert^2)^\alpha$, $\alpha > -1$. The Sobolev orthogonal projectors producing the approximations are with respect to an alternative yet equivalent inner product for the corresponding uniformly weighted $\mathrm{H}^1$ space. In order to obtain our approximation bounds, we study the orthogonal polynomial structure of this alternative Sobolev inner product obtaining, among other results, a characterization of its orthogonal polynomials as solutions of certain Sturm-Liouville problems. We do not rely on any particular basis of orthogonal polynomials, which allows for a streamlined and dimension-independent exposition.