Ultradifferentiable functions via the Laguerre operator

We define and characterize ultradifferentiable functions and their corresponding ultradistributions on $\RR^d_+$ using iterates of the Laguerre operator. The characterization is based on decay or growth conditions of the coefficients in their Laguerre series expansion. We apply our results to establish an isomorphism between subspaces of Pilipovi\'c spaces on $\RR^d$, and the spaces of ultradifferentiable functions on $\RR^d_+$.