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A Paley-Wiener theorem for generalized entire functions on infinite-dimensional spaces. (English. Russian original) Zbl 1029.46054
Izv. Math. 65, No. 2, 403-424 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, No. 2, 201-224 (2001).
The authors study the analogue of the Paley-Wiener theorem in the infinite dimensional context, using the space of Gâteaux analytic functions \(H_G(X)\) on a complex locally convex space \(X\) that have some additional boundedness properties. That such a theorem might be true in any situation is of interest, given the counterexample by S. Dineen and L. Nachbin [Israel J. Math. 13 (1972), 321-326 (1973; Zbl 0253.46097)].
Let \(\Theta\) be a collection of bounded subsets of \(X,\) and let \(A_\Theta(X) \equiv \{f \in H_G(X)\mid f\) is bounded on each \(B \in \Theta \}.\) For reasonable \(\Theta,\) \(A_\Theta(X)\) is complete, when given the locally convex topology of uniform convergence on sets \(B \in \Theta.\) For a vector space \(Y \subset X^\prime\) paired with \(X,\) let \(\text{Exp}_\Theta(Y) \equiv \{ \phi \in H_G(Y)\mid {\text{ for some }} M > 0 {\text{ and }} B \in \Theta, {\text{ we have }}|\phi(y)|\leq M~\exp(\sup_{y \in B}|\langle x,y\rangle|)\}.\) Motivated by the finite dimensional Fourier transform, the authors define the following: For \(\lambda \in A_\Theta^\prime(X)\) and \(y \in Y,\) let \({\mathcal F}(\lambda)(y) \equiv \langle\lambda, \exp\langle \cdot,y\rangle\rangle.\) The authors prove a type of Paley-Wiener theorem by showing that with some additional assumptions, \({\mathcal F}:A_\Theta^\prime(X) \to \text{Exp}_\Theta(Y)\) is a surjective strict morphism. Applications are given to sequence spaces \(X\) and \(Y.\)

46G20 Infinite-dimensional holomorphy
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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