  A class of translation-invariant Banach spaces of quasianalytic
ultradistributions is introduced and studied. They are Banach modules over a
Beurling algebra. Based on this class of Banach spaces, we define corresponding
test function spaces $\mathcal{D}^*_E$ and their strong duals
$\mathcal{D}'^*_{E'_{\ast}}$ of quasianalytic type, and study convolution and
multiplicative products on $\mathcal{D}'^*_{E'_{\ast}}$. These new spaces
generalize previous works about translation-invariant spaces of tempered
(non-quasianalytic ultra-) distributions; in particular, our new considerations
apply to the settings of Fourier hyperfunctions and ultrahyperfunctions. New
weighted $\mathcal{D}'^{\ast}_{L^{p}_{\eta}}$ spaces of quasianalytic
ultradistributions are analyzed.
