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oai:docta.ucm.es:20.500.14352/646812024-07-22T13:33:39Zcom_20.500.14352_14col_20.500.14352_15

The Paley-Wiener-Schwartz isomorphism in nuclear spaces Martínez Ansemil, José María Colombeau, J. F. 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]. 2023-06-21T02:02:28Z 2023-06-21T02:02:28Z 1981 journal article 0035-3965 https://hdl.handle.net/20.500.14352/64681 http://csm.ro/reviste/Revue_Mathematique/home_page.html http://www.ear.ro/index.php metadata only access Ed. Acad. Române