Abstract
In this paper, weight subspaces of the space of analytic functions on a bounded convex domain of the complex plane are considered. Descriptions of spaces that are strongly dual to the inductive and projective limits of uniformly weight spaces of analytic functions in a bounded convex domain D ⊂ ℂ are obtained in terms of the Fourier–Laplace transform. For each normed, uniformly weight space H(D, u), we construct the minimal vector space ℋi(D, u) containing H(D, u) and invariant under differentiation and the maximal vector space ℋp(D, u) contained in H(D, u) and invariant under differentiation. We introduce natural locally convex topologies on these spaces and describe strongly dual spaces in terms of the Fourier–Laplace transform. The existence of representing exponential systems in the space ℋi(D, u) is proved.
Similar content being viewed by others
References
N. F. Abuzyarova and R. S. Yulmukhametov, “Duals to weighted spaces of analytic functions,” Sib. Mat. Zh., 42, No. 1, 3–17 (2001).
L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley, New York (1970).
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo (2009).
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 2: Differential Operators with Constant Coefficients, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo (2004).
K. P. Isaev, K. V. Trounov, and R. S. Yulmukhametov, “Representation of functions in locally convex subspaces of A∞(D) by series of exponents,” Ufim. Mat. Zh., 9, No. 3, 50–62 (2017).
Yu. F. Korobeinik, “Representing systems,” Usp. Mat. Nauk, 36, No. 1 (217), 73–126 (1981).
A. F. Leontiev, Series of Exponents [in Russian], Nauka, Moscow (1976).
V. V. Napalkov, “On discrete weakly sufficient sets in some spaces of entire functions,” Izv. Akad. Nauk SSSR. Ser. Mat., 45, No. 5, 1088–1099 (1981).
V. V. Napalkov, “On comparison of topologies in certain spaces of entire functions,” Dokl. Akad. Nauk SSSR, 264, No. 4, 827–830 (1982).
V. V. Napalkov, “Spaces of analytic functions with given growth near the boundary,” Izv. Akad. Nauk SSSR. Ser. Mat., 51, No. 2, 287–305 (1987).
A. P. Robertson and W. Robertson, Topological Vector Spaces, Cambridge Univ. Press, Cambridge (1964).
R. S. Yulmukhametov, “Sufficient sets in a certain space of entire functions,” Mat. Sb., 116 (158), No. 3 (11), 427–439 (1981).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 153, Complex Analysis, 2018.
Rights and permissions
About this article
Cite this article
Bashmakov, R.A., Isaev, K.P. & Yulmukhametov, R.S. Representing Systems of Exponents in Weight Subspaces H (D). J Math Sci 252, 302–318 (2021). https://doi.org/10.1007/s10958-020-05162-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-05162-9