Analysis reveals asymptotic behavior of multivariate homogeneous polynomials on convex bodies, indicating optimal discretization methods may improve polynomial approximations.
In this paper we will study various extremal problems related to multivariate homogeneous polynomials on convex bodies. We will construct certain homogeneous needle polynomials on convex bodies which will play a central role in the study of the asymptotics of Christoffel functions and Lᵖ L p Markov type estimates for homogeneous polynomials. These new results will be applied in order to derive asymptotically optimal Marcinkiewicz-Zygmund type meshes and cubature formulas for homogeneous polynomials on convex bodies
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Kroó et al. (2025) studied this question.
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