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

Official URL: http://csm.ro/reviste/Revue_Mathematique/home_page.html

## 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 |
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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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