The weighted b-adic diaphony

The author introduces a new numerical measure for uniform distribution of sequences in [0,1)s, called weighted b-adic diaphony. It is proved that the computing complexity of the weighted b-adic diaphony of an arbitrary net, composed of N points in [0,1)s, is O(sN2). As special cases of the weighted b-adic diaphony we obtain some well-known kinds of the diaphony. An analogy of the inequality of Erdös–Turan–Koksma is given. We introduce the notion of limiting weighted b-adic diaphony, based on the Walsh functional system over finite groups as a characteristic of the behaviour of point nets in [0,1)∞. A general lower bound of the limiting weighted b-adic diaphony of an arbitrary net of N points in [0,1)∞ is proved. We introduce a class of weighted Hilbert space and prove a connection between the worst-case error of the integration of this space and the weighted b-adic diaphony.