An infinite dimensional version of the Paley-Wiener-Schwartz isomorphism
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In the infinite dimensional case it is known from a Dineen-Nachbin counterexample that the natural generalization of the Paley-Wiener-Schwartz theorem is no longer valid: The image through the Fourier transform F of the space ℰ′(E) is made of entire functions on E′ that, besides the usual inequality, satisfy a more technical condition. We define and study here a dense subspace, denoted by E(E), of ℰ(E) with a proper topology such that F (E′(E)) is made of the entire functions on E′ that satisfy only the classical inequality. For this reason this space E(E) is suited for the study of convolution and partial differential equations in spaces of C∞ functions on locally convex spaces.
KeywordsEntire Function Separable Hilbert Space Compact Open Topology Nuclear Space Topological Isomorphism
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- J. M. Ansemil and B. Perrot, On some linear partial difference differential equations in spaces of C ∞ functions defined on locally convexe spaces-preprint.Google Scholar
- P. J. Boland, Malgrange theorem for entire functions on nuclear spaces. Proceedings on Inf. Dim. Holo. Lectures Notes in Math. n∘ 364 (Springer) pp. 135–144.Google Scholar
- P. J. Boland, Holomorphic functions on nuclear spaces. Publicaciones del Departamento de Analisis Matematico. Universidad de Santiago de Compostela (1976).Google Scholar
- J. F. Colombeau, Sur les applications G-analytiques et analytiques en dimension infinie. Seminaire P. Lelong 1971–72 Lecture Notes in Math. n∘332 (Springer) 1973. pp. 48–58.Google Scholar
- J. F. Colombeau and B. Perrot, The Fourier Borel transform in infinitely many dimensions and applications. Functional Analysis, Holomorphy and Approximation Theory (editor S. Machado) Lecture Notes in Math. n∘ 843 Springer-1981, pp. 163–186.Google Scholar
- L. Nachbin, Lectures on the theory of distributions. Textos de Matematica Institute de Fisica e Matematica-Universidade do Recife 1964. Reprinted by University Microfilms International (USA) 1980Google Scholar