$\Omega$-ultradistributions

We develop a generalization of Beurling's approach to the construction of ultradistribution theory in which Fourier transformation is a basic tool. We establish a structure theorem on the representation of ultradistributions and a theorem of Paley–Wiener–Schwartz type. We illustrate the key role of extending the weights determining the spaces from $N$-dimensional real space, on which they are originally defined, to $N$-dimensional complex space.