Tempered ultradistributions as boundary values of analytic functions

Abstract

The spaces S a k {S_{{a_k}}} , S b q {S^{{b_q}}} and S a k b q S_{{a_{k}}}^{{b_q}} were introduced by I. M. Gel’fand as a generalization of the test function spaces of type S S ; the elements of the corresponding dual spaces are called tempered ultradistributions. It is shown that a function which is analytic in a tubular radial domain and satisfies a certain nonpolynomial growth condition has a distributional boundary value in the weak topology of the tempered ultradistribution space ( S b k a q ) ′ (S_{{b_{k}}}^{{a_{q}}})\prime , which is the space of Fourier transforms of elements in ( S a k b q ) ′ (S_{{a_{k}}}^{{b_{q}}})\prime . This gives rise to a representation of the Fourier transform of an element U ∈ ( S a k b q ) ′ U \in (S_{{a_{k}}}^{{b_{q}}})\prime having support in a certain convex set as a weak limit of the analytic function. Converse results are also obtained. These generalized Paley-Wiener-Schwartz theorems are established by means of a number of new lemmas concerning S a k b q S_{{a_{k}}}^{{b_{q}}} and its dual. Finally, in the appendix the equality S a k b q = S a k ∩ S b q S_{{a_k}}^{{b_q}} = {S_{{a_k}}} \cap {S^{{b_q}}} is proved.