The hybrid spectral test

The starting point of this paper is the interplay between the construction principle of a sequence and the characters of the compact abelian group that underlies the construction. In case of the Halton sequence in base $\mathbf b=(b_1, \ldots, b_s)$inthe$s$-dimensionalunitcube$[0,1)^s$, which is an important type of a digital sequence, this kind of duality principle leads to the so-called $\mathbf{b}$-adic function system and provides the basis for the $\mathbf{b}$-adic method, which we present in connection with hybrid sequences. This method employs structural properties of the compact group of $\mathbf b$-adicintegersaswellas$\mathbf b$-adic arithmetic to derive tools for the analysis of the uniform distribution of sequences in $[0,1{)}^s$. We first clarify the point which function systems are needed to analyze digital sequences. Then, we present the hybrid spectral test in terms of trigonometric-, Walsh-, and $\mathbf{b}$-adic functions. Various notions of diaphony as well as many figures of merit for rank-1 quadrature rules in Quasi-Monte Carlo integration and for certain linear types of pseudo-random number generators are included in this measure of uniform distribution. Further, discrepancy may be approximated arbitrarily close by suitable versions of the spectral test.