We describe spaces that are dual to some subspaces of the space C∞(D) relative to the inductive limits, where inductive limits, where \(D\subset {\mathbb{R}}^{p}\) is a bounded convex domain. For any logarithmically convex space of positive numbers \(\mathcal{M}=\left\{{M}_{k},k\in {\mathbb{Z}}_{+}^{p}\right\}\) we introduce the normed space \(C\left(D,\mathcal{M}\right)\) of functions \(f\in {C}^{\infty }\left(D\right)\) and prove that the Fourier-Laplace transform establishes a topological isomorphism between the strongly dual space and the projective limit of the normed spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley, New York etc. (1970).
B. A. Taylor, “Analytically uniform spaces of infinitely differentiable functions,” Commun. Pure Appl. Math. 24, No. 1, 39–51 (1971).
R. W. Braun, R. Meise, and B. A. Taylor, “Ultradifferentiable functions and Fourier analysis,” Result. Math. 17, No. 3-4, 206–237 (1990).
A. V. Abanin and I. A. Filip’ev, “Analytic implementation of the duals of some spaces of infinitely differentiable functions,” Sib. Math. J. 47, No 3. 397–409 (2006).
I. Kh. Musin and P. V. Yakovleva, “On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn,” Cent. Eur. J. Math. 10, No. 2. 665–692 (2012).
I. Kh. Musin, “On a space of entire functions rapidly decreasing on ℝn and its Fourier transform,” Concr. Oper. No 2, 120–138 (2015).
I. Kh. Musin, “On the Fourier–Laplace transform of functionals on a space of infinitely differentiable functions on a convex compact,” J. Math. Anal. Appl. 505, No 2, Article ID 125509 (2022).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 126, 2024, pp. 27-38.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lutsenko, A.V., Musin, I.K. & Yulmukhametov, R.S. Description of the Duals of Subspaces of Infinitely Differentiable Functions. J Math Sci 279, 478–492 (2024). https://doi.org/10.1007/s10958-024-07027-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07027-x