Tempered Ultradistributions as Boundary Values of Analytic Functions

Abstract

The spaces ${S_{{a_k}}}$, ${S^{{b_q}}}$ and $S_{{a_{k}}}^{{b_q}}$ were introduced by I. M. Gel’fand as a generalization of the test function spaces of type $S$; 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 $(S_{{b_{k}}}^{{a_{q}}})\prime$, which is the space of Fourier transforms of elements in $(S_{{a_{k}}}^{{b_{q}}})\prime$. This gives rise to a representation of the Fourier transform of an element $U \in (S_{{a_{k}}}^{{b_{q}}})\prime$ 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 $S_{{a_{k}}}^{{b_{q}}}$ and its dual. Finally, in the appendix the equality $S_{{a_k}}^{{b_q}} = {S_{{a_k}}} \cap {S^{{b_q}}}$ is proved.