We study approximation properties of weighted ${\mathrm{L}}^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form ${(1-{\Vert x\Vert }^2)}^{\alpha }{\prod }_{i=1}^d{\left|x_i\right|}^{{\gamma }_i}$, $\alpha ,{\gamma }_1,\dots ,{\gamma }_d>-1$. Said properties are measured in Dunkl–Sobolev-type norms in which the same weighted ${\mathrm{L}}^2$ norm is used to control all the involved differential–difference Dunkl operators, such as those appearing in the Sturm–Liouville characterization of similarly weighted ${\mathrm{L}}^2$-orthogonal polynomials, as opposed to the partial derivatives of Sobolev-type norms.

中文翻译：

---

球的Dunkl–Sobolev范数中的正交多项式投影误差

我们研究加权的近似性质 ${\mathrm{大号}}^2$正交投影仪投影到欧几里得单位球中有界度多项式的空间，其中权重为不变反射形式 ${（\mathrm{1个}-{\Vert X\Vert }^2）}^{\alpha }{\prod }_{\mathrm{一世}=\mathrm{1个}}^d{\left|X_{\mathrm{一世}}\right|}^{{\gamma }_{\mathrm{一世}}}$， $\alpha ，{\gamma }_{\mathrm{1个}}，\dots ，{\gamma }_d>-\mathrm{1个}$。所述属性以Dunkl-Sobolev型规范衡量，其中相同的权重${\mathrm{大号}}^2$ 范数用于控制所有涉及的微分-差分Dunkl算子，例如在Sturm-Liouville表征相似加权的算子中出现的算子 ${\mathrm{大号}}^2$-正交多项式，与Sobolev型规范的偏导数相反。