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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Tempered ultradistributions as boundary values of analytic functions

Author: R. S. Pathak
Journal: Trans. Amer. Math. Soc. 286 (1984), 537-556
MSC: Primary 46F20; Secondary 32A40
MathSciNet review: 760974
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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.

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Article copyright: © Copyright 1984 American Mathematical Society