  The aim of dynamical decoupling consists in the suppression of decoherence by
appropriate coherent control of a quantum register. Effectively, the
interaction with the environment is reduced. In particular, a sequence of $\pi$
pulses is considered. Here we present exact results on the suppression of the
coupling of a quantum bit to its environment by optimized sequences of $\pi$
pulses. The effect of various cutoffs of the spectral density of the
environment is investigated. As a result we show that the harder the cutoff is
the better an optimized pulse sequence can deal with it. For cutoffs which are
neither completely hard nor very soft we advocate iterated optimized sequences.
