C^∞ functions in infinite dimension and linear partial differential-difference equations with constant coefficients

Abstract

J.-F. Colombeau and S. Ponte [same journal 5 (1982), no. 2, 123–135;] defined and studied a dense linear subspace E(E) (with a stronger complete locally convex topology) of the space E(E) of all (Silva) C^∞-functions on a real nuclear bornological vector space E which is separated by its dual. The main point was that the "natural generalization'' of the classical Paley-Wiener-Schwartz theorem holds for the Fourier transform of E′(E) (but not for E′(E)). The authors of the present paper study (nonzero) linear partial differential-difference operators D with constant coefficients on E(E). If E is a real (DFN)-space, they prove that D is a surjective continuous operator on E(E) and that each solution u∈E(E) of Du=0 is the limit in E(E) of exponential polynomial solutions. Just like these results, also the general method of proof is similar to the one used in the classical case: The equation Df=g, g∈E(E), is transposed and, using Fourier transforms, the proof of the surjectivity of D is reduced to a division theorem. (Here the main results in the article of Colombeau and Ponte [op. cit.], as well as some results of L. Ehrenpreis, are used; of course, the Hahn-Banach theorem is needed, too.) Finally, the authors show an extension theorem analogous to the one (due to P. J. Boland ) in the holomorphic case; viz., for a closed subspace F of a real (DFN)-space E, the restriction map from E(E) to E(F) is surjective.