Orthogonal polynomial projection error in Dunkl–Sobolev norms in the ball

We study approximation properties of weighted $\mathrm{L}^2$-orthogonal projectors onto spaces polynomials bounded degree in the Euclidean unit ball, where weight is reflection-invariant form $(1-\lVert x \rVert^2)^\alpha \prod_{i=1}^d \lvert x_i \rvert^{\gamma_i}$, $\alpha, \gamma_1, \dots, \gamma_d > -1$. Said are measured Dunkl-Sobolev-type norms which same $\mathrm{L}^2$ norm used to control all involved differential-difference Dunkl operators, such as those appearing Sturm-Liouville characterization similarly polynomials, opposed partial derivatives Sobolev-type norms. The method proof relies on instead bases orthogonal greatly simplifies exposition.