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 ℤd-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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Brandolini, L., Colzani, L., Gigante, G., Travaglini, G.: On the Koksma–Hlawka inequality. J. Complex, submitted
Harman, G.: Variations on the Koksma–Hlawka inequality. Unif. Distrib. Theory 5, 65–78 (2010)
Hickernell, F.J.: Koksma–Hlawka inequality. In: Kotz, S., Read, C.B., Banks, D.L. (eds.): Encyclopedia of Statistical Sciences. Wiley-Interscience, New York (2006)
Hlawka, E.: The Theory of Uniform Distribution. A B Academic Publishers, Berkhamsted (1984)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Dover, New York (2006)
Matoušek, J.: Geometric Discrepancy: An Illustrated Guide. Springer, Berlin (2010)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
Zaremba, S.K.: Some applications of multidimensional integration by parts. Ann. Pol. Math. 21, 85–96 (1968)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Italia
About this chapter
Cite this chapter
Brandolini, L., Colzani, L., Gigante, G., Travaglini, G. (2013). A Koksma–Hlawka Inequality for Simplices. In: Picardello, M. (eds) Trends in Harmonic Analysis. Springer INdAM Series, vol 3. Springer, Milano. https://doi.org/10.1007/978-88-470-2853-1_3
Download citation
DOI: https://doi.org/10.1007/978-88-470-2853-1_3
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2852-4
Online ISBN: 978-88-470-2853-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)