Diaphony, discrepancy, spectral test and worst-case error

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. Further we prove that the diaphony in base 2 coincides, for a special choice of weights, 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 L2 discrepancy of randomly digitally shifted points.