Resumen:
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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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