The weighted b-adic diaphony

Abstract

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 ( sN 2 ) . 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.