  Nowadays the theory and application of wavelet decompositions plays an
important role not only for the study of function spaces (of Lebesgue, Hardy,
Sobolev, Besov, Triebel-Lizorkin type) but also for its applications in signal
and numerical analysis, partial differential equations and image processing. In
this context it it a hard problem to construct wavelet bases for suitable
function spaces on domains, e. g. the unit cube. A big step in this direction
are the contributions of Hans Triebel from 2006 to 2008 where he constructed
Riesz bases for classes of Besov- and Triebel-Lizorkin spaces on domains,
starting with Daubechies wavelets. But there was a problem coming from the
method: He had to exclude a big number of function spaces, in particular a
large class of classical Sobolev spaces. The main goal of this thesis is a
construction of Riesz bases of wavelet systems also for the exceptional cases
using a modification of the function spaces - the so-called reinforced function
spaces.
