# Weighted ℓ_q approximation problems on the ball and on the sphere

Let L_q,μ, 1≤ q<∞, μ≥0, denote the weighted L_q space with the classical Jacobi weight w_μ on the ball B^d. We consider the weighted least ℓ_q approximation problem for a given L_q,μ-Marcinkiewicz-Zygmund family on B^d. We obtain the weighted least ℓ_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 ℓ_q approximation theorem and the least squares quadrature errors on the sphere.

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