We introduce a normed space of functions, holomorphic in a bounded convex domain. Its elements are infinitely differentiable up to the boundary, and all their derivatives satisfy estimates specified by a convex sequence of positive numbers. We consider its largest linear subspace that is invariant with respect to the operator of differentiation and provide it with the natural topology of projective limit. We establish duality between this subspace and some space of entire functions. Based on this, we construct a representing system of exponentials in the subspace.
analytic function weighted space locally convex space sufficient set representing system of exponentials
This is a preview of subscription content, log in to check access.
Leontiev, A. F. Series of Exponentials (Nauka, Moscow, 1976) [in Russian].Google Scholar
Levin, B. Ja., Lyubarskii, Yu. I. “Interpolation by Means of Special Classes of Entire Functions and Related Expansions in Series of Exponentials”, Mathematics of the USSR-Izvestiya 9, No. 3, 621–662 (1975).CrossRefGoogle Scholar
Isaev, K. P. “Riesz Bases of Exponents in Bergman Spaces on Convex Polygons”, Ufimsk. Mat. Zh. 2, No. 1, 71–86 (2010) [in Russian].zbMATHGoogle Scholar
Lyubarskii, Yu. I. “Exponential Series in Smirnov Spaces and Interpolation by Entire Functions of Special Classes”, Mathematics of the USSR-Izvestiya 32, No. 3, 563–586 (1989).MathSciNetCrossRefGoogle Scholar
Ehrenpreis, L. Fourier Analysis in Several Complex Variables (Willey Interscience publishers, New York, 1970).zbMATHGoogle Scholar
Napalkov, V. V. “Spaces of Analytic Functions of Prescribed Growth Near the Boundary”, Math. USSRIzv. 30, No. 2, 263–281 (1988).zbMATHGoogle Scholar
Abanin, A. V. “Characterization of Minimal Systems of Exponents of Representative Systems of Generalized Exponentials”, Soviet Mathematics 35, No. 2, 1–12 (1991).MathSciNetzbMATHGoogle Scholar
Abanin, A. V. “Absolutely Representing Systems of Exponentials of Minimal Type in Spaces of Functions With Prescribed Growth Near the Boundary”, Russian Mathematics 37, No. 10, 72–75 (1993).MathSciNetzbMATHGoogle Scholar
Abanin, A. V., Le Hai Khoi, Nalbandyan, Yu. S. “Minimal Absolutely Representing Systems of Exponentials for A−∞(Ω)”, J. Approx. Theory 163, No. 10, 1534–1545 (2011).MathSciNetCrossRefzbMATHGoogle Scholar