We describe local and global properties of wavelet transforms of
ultradifferentiable functions. The results are given in the form of continuity
properties of the wavelet transform on Gelfand-Shilov type spaces and their
duals. In particular, we introduce a new family of highly time-scale localized
spaces on the upper half-space. We study the wavelet synthesis operator (the
left-inverse of the wavelet transform) and obtain the resolution of identity
(Calder\'{o}n reproducing formula) in the context of ultradistributions
