%0 Journal Article
%A Martínez Ansemil, José María
%A Colombeau, J. F.
%T The Paley-Wiener-Schwartz isomorphism in nuclear spaces
%D 1981
%@ 0035-3965
%U https://hdl.handle.net/20.500.14352/64681
%X The authors are concerned with the characterization of those functions holomorphic on EC′ which are Fourier transforms of elements of ′ (E). Here E is a complete bornological vector space over R,  (E) stands for the space of all complex-valued C∞ -functions on E, and EC denotes the complexification and E′ the (bornological) dual of E. The authors start with carrying over the classical Paley-Wiener-Schwartz theorem from RN to vector spaces E which have finite-dimensional bornology. (The only important infinite-dimensional member of this class seems to be ⊕NR, the space of finite sequences.) Then they show that the counterexample of S. Dineen and L. Nachbin [Israel J. Math. 13 (1972), 321–326 (1973)] extends to all vector spaces which possess an infinite-dimensional bounded set, i.e., the Paley-Wiener-Schwartz condition (PWS) does not give the desired characterization in most cases. Finally they formulate a further condition A and they prove that a function holomorphic on EC′ is the Fourier transform of an element of E′ (E) if and only if it satisfies PWS and A, provided E is endowed with a nuclear bornology. For Banach spaces E, a similar result was obtained by T. Abuabara earlier [Advances in holomorphy (Rio de Janeiro, 1977), pp. 1–29, North-Holland, Amsterdam, 1979].
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