A. M. Sobina^{1, 2}

¹Ufa State Petroleum Technological University (Ufa, Russian Federation)

²Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. In the paper we introduce and study a new generalization of the classical Laplace transform for functionals defined on spaces of entire functions of several complex variables. The kernel of the transform is chosen to be an entire function of a special form, which naturally generalizes the exponential function and has applications in problems related to differential equations with generalized partial derivatives, as well as in the theory of multidimensional analogues of Fock spaces. One of the key problems in the theory of functions of several complex variables is the description of growth of entire functions. Various growth characteristics of functions may be used for this purpose, such as conjugate orders and types, orders and types with respect to each variable, and others. In this work, for the function defining the kernel of the generalized Laplace transform, we compute orders and types with respect to each variable, which makes it possible to characterize the function's behaviour at infinity. Furthermore, we establish an isomorphism between the space dual to the space of entire functions of several complex variables, endowed with the topology of compact convergence, and the space of functions that are analytic in each variable outside some polydisk. This result plays an important role in subsequent constructions. On its basis, we prove that the introduced generalized Laplace transform realizes an isomorphism between the space of continuous linear functionals over the space of entire functions of several complex variables and the space of entire functions with a prescribed order of growth. Thus, the proposed approach opens up new possibilities for the study of function spaces and their applications to differential equations.

Key Words: Entire functions of several complex variables, the orders and types of an entire function with respect to each variable, generalized Laplace transform

For citation: A. M. Sobina. On a generalization of the Laplace transform. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:3(2026), 193–211. DOI: https://doi.org/10.15507/2079-6900.28.202603.193-211

Information about the author:

Aliya M. Sobina, Senior Lecturer at the Department of Information Technologies and Applied Mathematics, Ufa State Petroleum Technological University (1 Kosmonavtov St., Ufa 450064, Russia), Postgraduate Student, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: http://orcid.org/0000-0002-9014-9508, aliya-0887@mail.ru