Given a sequence of Marcinkiewicz–Zygmund inequalities in L2\documentclass[12pt]{minimal}
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\begin{document}$$L_2$$\end{document} on a compact space, Gröchenig (J Approx Theory 257:105455, 2020) discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all 1≤p≤∞\documentclass[12pt]{minimal}
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\begin{document}$$1\le p\le \infty $$\end{document}, we develop weighted least ℓp\documentclass[12pt]{minimal}
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\begin{document}$$\ell _p$$\end{document} approximation induced by a sequence of Marcinkiewicz–Zygmund inequalities in Lp\documentclass[12pt]{minimal}
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\begin{document}$$L_p$$\end{document} on a compact smooth Riemannian manifold M\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb M$$\end{document} with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in Lq,1≤q≤∞\documentclass[12pt]{minimal}
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\begin{document}$$L_q,\,1\le q\le \infty $$\end{document}, and least quadrature errors for both Sobolev spaces Hpr(M),r>d/p\documentclass[12pt]{minimal}
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\begin{document}$$H_p^r(\mathbb M), \, r>d/p$$\end{document} generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces Bp,γr(M),0<γ≤∞,r>d/p\documentclass[12pt]{minimal}
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\begin{document}$$B_{p,\gamma }^r(\mathbb M),\, 0<\gamma \le \infty ,\, r>d/p $$\end{document} defined by best ”polynomial” approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces.
