We estimate the error in the approximation of the integral of a smooth function over a parallelepiped Ω or a simplex S by Riemann sums with deterministic ℤ d -periodic nodes. These estimates are in the spirit of the Koksma–Hlawka inequality, and depend on a quantitative evaluation of the uniform distribution of the sampling points, as well as on the total variation of the function. The sets used to compute the discrepancy of the nodes are parallelepipeds with edges parallel to the edges of Ω or S. Similarly, the total variation depends only on the derivatives of the function along directions parallel to the edges of Ω or S.
Brandolini, L., Colzani, L., Gigante, G., Travaglini, G. (2013). A Koksma–Hlawka Inequality for Simplices. In M. Picardello (a cura di), Trends in Harmonic Analysis (pp. 33-46). Springer International Publishing [10.1007/978-88-470-2853-1_3].
A Koksma–Hlawka Inequality for Simplices
COLZANI, LEONARDO;TRAVAGLINI, GIANCARLO
2013
Abstract
We estimate the error in the approximation of the integral of a smooth function over a parallelepiped Ω or a simplex S by Riemann sums with deterministic ℤ d -periodic nodes. These estimates are in the spirit of the Koksma–Hlawka inequality, and depend on a quantitative evaluation of the uniform distribution of the sampling points, as well as on the total variation of the function. The sets used to compute the discrepancy of the nodes are parallelepipeds with edges parallel to the edges of Ω or S. Similarly, the total variation depends only on the derivatives of the function along directions parallel to the edges of Ω or S.
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