Discrepancy and numerical integration on metric measure spaces

Abstract

We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz-Zygmund inequality and ad hoc definitions of function spaces on metric measure spaces. The same techniques are used to prove the existence of point distributions in metric measure spaces with small $L^p$ discrepancy with respect to certain classes of subsets, for example metric balls.

References in corpus (4)

• QMC designs: optimal order Quasi Monte Carlo Integration schemes on the sphere
• Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces
• Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces
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Cited by in corpus (2)

• Optimal asymptotic bounds for designs on manifolds
• Stolarsky's invariance principle for projective spaces