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Weighted $\ell_q$ approximation problems on the ball and on the sphere

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.