Polyelement functional equations related to the Carleman kernel and their applications

The review article considers the properties and applications of polyelement functional equations related to the T. Carleman kernel. Various results obtained in such sections of mathematics as the theory of entire functions, convolution operators, automorphic functions, etc. are described. In the work, the method of integral equations is used, which is based on the representation of the solution in the form of a Cauchy-type integral with an unknown density. The focus of the study of various difference and sum equations in the class of functions that are holomorphic in some neighborhood of a point at infinity and vanish there. We also consider interpolation problems for entire functions of exponential type from the class A.