We study approximation properties of weighted $L^2$-orthogonal projectorsonto spaces of polynomials of bounded degree in the Euclidean unit ball, wherethe weight is of the generalized Gegenbauer form $x \mapsto(1-\|x\|^2)^\alpha$, $\alpha > -1$. Said properties are measured inSobolev-type norms in which the same weighted $L^2$ norm is used to control allthe involved weak derivatives. The method of proof does not rely on anyparticular basis of orthogonal polynomials, which allows for a short,streamlined and dimension-independent exposition.
展开▼
机译：我们研究加权的$ L ^ 2 $-正交投影机对欧几里得单位球中有界度多项式的空间的逼近性质，其中权重是广义Gegenbauer形式的$ x \ mapsto（1- \ | x \ | ^ 2）^ \ alpha $，$ \ alpha> -1 $。所述性质是在Sobolev型范数中测量的，其中相同的加权$ L ^ 2 $范数用于控制所有涉及的弱导数。证明方法不依赖于正交多项式的任何特定基础，从而可以进行简短，流线型和尺寸无关的说明。
展开▼
