APPROXIMATION OF THE DISCONTINUOUS FUNCTION OF TWO VARIABLES BY DISCONTINUOUS INTERLINATION SPLINES USING TRIANGULAR ELEMENTS

Abstract

The work is devoted to the development of a method for approximating discontinuous functions using the operator of interlination of two variables functions. These operators reconstruction functions (possibly approximately) from their known traces on a given line system. It is such experimental data that are used in remote remote methods, in particular, in computed tomography. That is, they provide an opportunity to build operators whose integrals over the indicated lines (linear integrals) will be equal to the integrals of the function being restored. So, interlination is a mathematical apparatus naturally associated with the task of restoring the characteristics of objects from their known projections. There are many practically important scientific and technical branches in which objects of research are mathematically described by values that have a discontinuity. Such objects often arise also in tasks using remote methods. To date, there is no general theory of descriptions of phenomena and processes described by discontinuous functions. The paper constructs and explores discontinuous interlination operators for approximating discontinuous functions of two variables according to its known traces (projections) on a system of lines using arbitrary triangular elements. On the basis of the created splineinterlinants, a method is constructed for approximating functions that have discontinuities of the first kind and whose domain of definition is divided into triangular elements. Moreover, the constructed discontinuous structures include, as a special case, the classic continuous interlination splines. The experimental data are one-sided traces of a function on a system of given lines; precisely such data are used in tomography. The paper presents theorems on interlining properties and errors of constructed discontinuous structures. The constructed approximation method allows us to approximate the discontinuous function, avoiding the Gibbs phenomenon. The examples confirming the effectiveness of the proposed method are considered. The proposed method for approximating discontinuous functions can be used for mathematical modeling of discontinuous processes in medical, geological, space and other studies.