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The Paley-Wiener-Schwartz isomorphism in nuclear spaces

Ansemil, José María M. and Colombeau, J.F. (1981) The Paley-Wiener-Schwartz isomorphism in nuclear spaces. Revue roumaine de mathematiques pures et appliquees, 26 (2). pp. 169-181. ISSN 0035-3965

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Abstract

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].

Item Type:Article
Uncontrolled Keywords:Paley-Wiener-Schwartz theorem; bornological dual; complete bornology; vector space of Silva C-infinity-functions; Silva holomorphic function; nuclear bornology; Fourier-Laplace transforms; growth property
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16844
Deposited On:24 Oct 2012 08:53
Last Modified:13 Nov 2013 15:25

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