  We enrich the class of power-constructible functions, introduced in [CCRS23],
to a class of algebras of functions which contains all complex powers of
subanalytic functions, their parametric Mellin and Fourier transforms, and
which is stable under parametric integration. By describing a set of generators
of a special prepared form we deduce information on the asymptotics and on the
loci of integrability of the functions of the class. We furthermore identify a
subclass which is the smallest class containing all power-constructible
functions and stable under parametric Fourier transforms and right-composition
with subanalytic maps. This subclass is also stable under parametric
integration, under taking pointwise and $L^p$limits, and under parametric
Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in
the power-logarithmic scale, uniformly in the parameters, for functions in this
subclass.
