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.
Categories: math.FA