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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 $$Y.$$

##### MSC:
 46G20 Infinite-dimensional holomorphy 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
##### Keywords:
Paley-Wiener theorem; infinite dimensional holomorphy
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