Lp-discrepancy and beyond of higher order digital sequences
Sprache des Vortragstitels:
MCQMC 2016, 12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Stanford, California, August 14-19, 2016
Sprache des Tagungstitel:
The Lp-discrepancy is a quantitative measure for the
irregularity of distribution modulo one of infinite sequences.
In 1986 Proinov proved for all p > 1 a lower
bound for the Lp-discrepancy of general infinite sequences
in the d-dimensional unit cube, but it remained
an open question whether this lower bound
is best possible in the order of magnitude until recently.
In 2014 Dick and Pillichshammer gave a first
construction of an infinite sequence whose order of
L2-discrepancy matches the lower bound of Proinov.
Here we give a complete solution to this problem
for all finite p > 1. We consider so-called order 2
digital (t; d)-sequences over the finite field with two
elements and show that such sequences achieve the
optimal order of Lp-discrepancy simultaneously for
all p 2 (1;1).
Beyond this result, we estimate the norm of the
discrepancy function of those sequences also in the
space of bounded mean oscillation, exponential Orlicz
spaces, Besov and Triebel-Lizorkin spaces and
give some corresponding lower bounds which show
that the obtained upper bounds are optimal in the
order of magnitude.
The talk is based on joint work with Josef Dick,
Lev Markhasin, and Friedrich Pillichshammer.