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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 in the -dimensional unit cube , which is an
important type of a digital sequence, this kind of duality principle leads to
the so-called -adic function system and provides the basis for the
-adic method, which we present in connection with hybrid sequences.
This method employs structural properties of the compact group of -adic integers as well as -adic arithmetic to derive tools for the
analysis of the uniform distribution of sequences in .
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 -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.Comment: This is the first version of a survey paper for the RICAM special
semester in fall 201