其他名称：
25-21-00062
项目负责人：
Melikhov Sergej
发表日期：
2025
主持机构：
Southern Federal University,
国家：
俄罗斯
开始日期：
2025
结束日期：
2026
简介：
It is supposed to study backward shift and Hadamard-type operators in spaces of holomorphic, infinitely differentiable and ultradifferentible functions. Operators of this type are of interest both in their own right and in connection with applications. The backward shift operator and, more generally, the Pommiez difference operator arose, in particular, in the construction of the interpolating function by A.F. Leontiev, which has numerous applications in the theory of convolution operators and in representations of functions by exponential series. It can be realized as the conjugate of the Volterra integration operator and is closely related to the Duhamel product. At present, Duhamel algebras are being studied quite intensively. The problems associated with the backward shift, in addition to those indicated, have applications in operational (operator) calculus, in the Cauchy problem, in the solving of the problem of the spectral multiplicity of a linear operator, in spectral theory in a generalized sense, in the sloping beach problem. Recently, the shifts associated with it have made it possible to introduce the multiplication in the (whole) space of generalized functions on an interval of the real line. For backward shift operators, quite a lot of results have been obtained for the spaces of holomorphic functions in a domain in the complex plane, in particular in the unit disk, and for spaces of entire functions of exponential type realizing duals to spaces of the previous type for one complex variable. Some important spaces in this direction have been studied little or not at all (for example, spaces of holomorphic functions of a given growth or a given boundary smoothness near the boundary of the analyticity domain). The case of several variables has been studied also muss less. The authors plan to consider the mentioned situations in detail. In this direction the cyclic vectors of the system of partial backward shift operators, their proper closed invariant subspaces, algebras with the multiplication associated with the system of backward shift operators will be studied, in particular, applications to the description of the ideals of algebras, multiplication in which is the multidimensional Duhamel product, will be obtained. It will be applied to the solution of Cauchy problems in spaces of holomorphic, infinitely differentiable and ultradifferentiable functions. The dual approach will make it possible to obtain corresponding results for the Volterra operator in conjugate spaces or their implementations.
Hadamard-type operators arose as a variation of the general mathematical concept of the componentwise multiplication, they originate in the theory of the Hadamard product of holomorphic functions. A special case of such operators are the Euler operators. There are a lot of works devoted to Hadamard operators, algebras of such operators in spaces of real-analytic, infinitely differentiable functions, in spaces of distributions. In spaces of holomorphic functions in domains they have been studied in detail only in the case of one complex variable. It is supposed to study in detail such operators in spaces of holomorphic functions in domains in a multidimensional complex space, to study modules and algebras with multiplicative and almost multiplicative multiplication, that is, with corresponding convolutions.
In the one-dimensional and in the multidimensional case for the theory of backward shift operators and Hadamard-type operators the spaces of entire functions that realize the conjugate to the spaces of infinitely differentiable and ultradifferentiable functions are of interest. They are currently being investigated in various situations. Interest in them is due to the fact that they include the Denjoy-Carleman and Gevrey classes, which are used in partial differential equations. For them, Whitney-type continuation theorems, Borel-type theorems on the interpolation are proved, and properties of differential operators are studied. The extremes in the scale of these spaces are the spaces of infinitely differentiable and real analytic or entire functions.Operators of the type indicated above will also be studied in such spaces, in the realizations of their duals.
专业领域：
信息技术
语种：
英语
