We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.
The classical Koksma Hlawka inequality does not apply to functions with simple discontinuities. Here we state a Koksma Hlawka type inequality which applies to piecewise smooth functions f χ Ω , with f smooth and Ω a Borel subset of [0, 1] d :
Abstract. Let B be a convex body in R 2 , with piecewise smooth boundary and let b B denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical L p -averages of b B and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average numberofinteger lattice points in large bodies randomly positioned in the plane.
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