Abstract
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is known from a Dineen-Nachbin counterexample that the natural generalization of the Paley-Wiener-Schwartz theorem is no longer valid: The image through the Fourier transform F of the space ℰ′(E) is made of entire functions on E′ that, besides the usual inequality, satisfy a more technical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.
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References
T. Abuabara, A version of the Paley-Wiener-Schwartz theorem in infinite dimension. Advances in Holomorphy (editor J. A. Barroso) North Holland Math. Studies 34 (1979), 1–30.
J. M. Ansemil and J. F. Colombeau, The Paley-Wiener-Schwartz isomorphism in nuclear spaces. Revue Roumaine de Math. Pures et Appliquées—tome 26 No 2, 1981, p. 169–181.
J. M. Ansemil and B. Perrot, On some linear partial difference differential equations in spaces of C ∞ functions defined on locally convexe spaces-preprint.
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.
P. J. Boland, Holomorphic functions on nuclear spaces. Publicaciones del Departamento de Analisis Matematico. Universidad de Santiago de Compostela (1976).
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.
J. F. Colombeau and B. Perrot, Reflexivity and kernels in infinite dimensional holomorphy. Portugaliae Mathematica, Vol 46, Fasc 3-4, (1977), pp. 291–300.
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.
S. Dineen and L. Nachbin, Entire functions of exponential type bounded on the real axis and Fourier transforms of distributions with bounded supports. Israel Journal of Mathematics, vol. 13 (1972), 321–326.
L. Nachbin, Sur les algébres denses de fonctions différentiables sur une variété. Comptes Rendus Acad. des Sciences. Paris 228 (1949), 1549–1551.
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) 1980
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Colombeau, J.F., Ponte, S. An infinite dimensional version of the Paley-Wiener-Schwartz isomorphism. Results. Math. 5, 123–135 (1982). https://doi.org/10.1007/BF03323309
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DOI: https://doi.org/10.1007/BF03323309