Martínez Ansemil, José MaríaColombeau, J. F.2023-06-212023-06-2119810035-3965https://hdl.handle.net/20.500.14352/64681The 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].journal articlehttp://csm.ro/reviste/Revue_Mathematique/home_page.htmlhttp://www.ear.ro/index.phpmetadata only access517.98Paley-Wiener-Schwartz theoremBornological dualComplete bornologyVector space of Silva C-infinity-functionsSilva holomorphic functionNuclear bornologyFourier-Laplace transformsGrowth propertyAnálisis funcional y teoría de operadores