Relationship between solutions of discrete inverse and auxiliary problems for a hyperbolic equation

One-dimensional and multidimensional methods for solving inverse problems for a hyperbolic equation by the Gelfand-Levitan method lead to the numerical solution of Fredholm integral equations of the first and second kind. This paper examines the connection between a discrete inverse problem and solutions to a discrete auxiliary problem for a hyperbolic equation studied by the Gelfand-Levitan method. First, we present the formulations of the discrete inverse problem and the discrete auxiliary problem for the hyperbolic equation, as well as the properties of the grid functions that are solutions to these problems. Lemmas are proved about the structure of the grid function determined by the solution of the auxiliary discrete problem, and the connection of this grid function with the desired grid function, which is the solution to the discrete inverse problem. When proving the lemmas, the properties of the solution to the discrete auxiliary problem and the discrete analogue of the Dirac delta function are taken into account. A theorem showing the existence of a discrete inverse problem solution and its uniqueness is proved.