论文标题
加权$ \ ell_q $ $近似问题在球和球体上
Weighted $\ell_q$ approximation problems on the ball and on the sphere
论文作者
论文摘要
令$ l_ {q，μ}，\，\，1 \ le q <\ infty，\μ\ ge0，$表示加权$ l_q $ space，带有经典的jacobi重量$w_μ$，$ \ bbb bbb bbb b^d $。我们考虑给定$ l_ {q，μ} $的加权$ \ ell_q $近似问题 - $ \ bbb b b^d $上的Marcinkiewicz-Zygmund family。我们获得了加权$ \ ell_q $近似错误的加权sobolev空间$ w_ {q，μ}^r $，$ r>（d+2μ）/q $，这是最佳订单。我们还讨论了$ l_ {2，μ} $ - marcinkiewicz-zygmund家族引起的最小二乘正交正交，并获得$ w_ {2，μ}^r $，$ r $，$ r>（d+2μ）/2的正交错误，也是最佳订单。同时，我们给出了对应的加权$ \ ell_q $近似定理，而最小二乘在球体上的正方形正交错误。
Let $L_{q,μ},\, 1\le q<\infty, \ μ\ge0,$ denote the weighted $L_q$ space with the classical Jacobi weight $w_μ$ on the ball $\Bbb B^d$. We consider the weighted least $\ell_q$ approximation problem for a given $L_{q,μ}$-Marcinkiewicz-Zygmund family on $\Bbb B^d$. We obtain the weighted least $\ell_q$ approximation errors for the weighted Sobolev space $W_{q,μ}^r$, $r>(d+2μ)/q$, which are order optimal. We also discuss the least squares quadrature induced by an $L_{2,μ}$-Marcinkiewicz-Zygmund family, and get the quadrature errors for $W_{2,μ}^r$, $r>(d+2μ)/2$, which are also order optimal. Meanwhile, we give the corresponding the weighted least $\ell_q$ approximation theorem and the least squares quadrature errors on the sphere.
