Abstract
In the Gevrey space A(q) of functions analytic in a nonconvex bounded region in C−, we consider a convolution operator L generated by some entire functiona. A necessary and sufficient condition under which the equality L(A(q))=A(q) holds is established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
O. V. Gorina, “Solvability of a class of differential equations in a space of functions analytic in a nonconvex region and having specified order of growth,” Differents. Urav.,26, No. 5, 760–769 (1990).
Yu. F. Korobeinik, “Existence of analytic solution of differential equation of infinite order and the nature of its domain of analyticity,” Mat. Sb.,80, No. 1, 52–76 (1969).
S. V. Znamenskii, Nontraditional Nonconvexity in the Sense of Planar Regions and Compacta and Properties of Holomorphic Solutions of Differential Equations of Infinite Order [in Russian], Krasnoyarsk (1980), dep. VINITI July 15, 1980, No. 3063-80.
I. F. Krasichkov-Ternovskii, “Invariant subspaces of analytic functions. II. Propagation of spectral synthesis,” Mat. Sb.,88, No. 3, 331–352 (1972).
B. A. Derzhavets, “Spaces of functions analytic in certain linear convex regions in Cn exhibiting specified behavior near the boundary,” Izv. Vuzov. Matem., No. 6, 10–13 (1985).
O. V. Epifanov, “Solvability of Cauchy-Riemann equation with restrictions on the growth of functions and weight approximation of analytic functions,” Izv. Vuzov. Matem., No. 2, 49–52 (1990).
R. Edwards, Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).
B. Ya. Levin, Distribution of the Roots of Entire Functions [in Russian], Fizmatgiz, Moscow (1956).
O. V. Gorina, “Solvability of differential equations with constant coefficients in classes of analytic functions with specified estimates of the derivatives,” in: Investigations in Theory of Approximations [in Russian], In-st. Matem., Computer Center, Ufa (1990), pp. 34–43.
Yu. F. Korobeinik and O. V. Gorina, On Interpolation Analysis in a Class of Entire Functions. Applications and Basis Property [in Russian], Rostov-on-Don (1986), dep. VINITI February 2, 1987, No. 744-B87.
O. V. Epivanov, “Multiplication operators in spaces of entire functions of finite order and convolution-type operators,” Mat. Sb.,120, No. 4, 505–527 (1983).
O. V. Epifanov, A Theorem on the Distribution of the Singular Points of Entire Functions of Finite Order and Its Applications [in Russian], Rostov-on-Don (1986), dep. VINITI September 1, 1986, No. 6328-B.
A. F. Leont'ev, Entire Functions. Exponential Series [in Russian], Nauka, Moscow (1983).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 35–43, September, 1992.
Rights and permissions
About this article
Cite this article
Gorina, O.V. Solvability of convolution equation in the Gevrey class of functions analytic in a nonconvex region. Math Notes 52, 895–902 (1992). https://doi.org/10.1007/BF01209608
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01209608