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.
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