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Diaphony, discrepancy, spectral test and worst-case error. (English) Zbl 1193.65003

Summary: In this paper various measures for the uniformity of distribution of a point set in the unit cube are studied. We show how the diaphony and spectral test based on Walsh functions appear naturally as the worst-case error of integration in certain Hilbert spaces which are based on Walsh functions. Furthermore, it has been shown that this worst-case error equals to the root mean square discrepancy of an Owen scrambled point set.
We also prove that the diaphony in base 2 coincides with the root mean square worst-case error for integration in certain weighted Sobolev spaces. This connection has also a geometrical interpretation, which leads to a geometrical interpretation of the diaphony in base 2. Furthermore we also establish a connection between the diaphony and the root mean square weighted \(\mathcal L_2\) discrepancy of randomly digitally shifted points.

MSC:

65C05 Monte Carlo methods
11K06 General theory of distribution modulo \(1\)
11K38 Irregularities of distribution, discrepancy
60E05 Probability distributions: general theory
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