  We study approximation properties generated by highly regular scaling
functions and orthonormal wavelets. These properties are conveniently described
in the framework of Gelfand-Shilov spaces. Important examples of
multiresolution analyses for which our results apply arise in particular from
Dziuba\'{n}ski-Hern\'{a}ndez construction of band-limited wavelets with
subexponential decay. Our results are twofold. Firstly, we obtain approximation
properties of multiresolution expansions of Gelfand-Shilov functions and
(ultra)distributions. Secondly, we establish convergence of wavelet series
expansions in the same regularity framework.
