CUBATURE FORMULA FOR APPROXIMATE CALCULATION INTEGRAL OF HIGHLY OSCILLATING FUNCTION OF TREE VARIABLES (IRREGULAR CASE)

Abstract

Context. The integrals of highly oscillating functions of many variables are one of the central concepts of digital signal and image processing. The object of research is a digital processing of signals and images using new information operators. Objective. The work aims to construct a cubature formula for the approximate calculation of the triple integral of a rapidly oscillating function of a general form.Method. Modern methods of digital signal processing are characterized by new approaches to obtaining, processing and analyzing information. There is a need to build mathematical models in which information can be given not only by the values of the function at points, but also as a set of traces of the function on the planes and as a set of traces of the function on the lines. There are algorithms which are optimal by accuracy for calculating the integrals of highly oscillating functions of many variables (regular case), which involve different types of information in their construction. As a solution of a broader problem for the irregular case, the work presents the cubature formula for the approximate calculation of the triple integral of the highly oscillating function in a general case. The presented algorithm for approximate calculation of the integral is based on the application of operators that restore the function of three variables using a set of traces of functions on the mutually perpendicular planes. Operators use piece-wise splines as auxiliary functions. The cubature formula correlates with a formula of the Filon type. An error estimation of the approximation of the integral from the highly oscillating function by the cubature formula on the class of differential functions is obtained.Results. The cubature formula of the approximate calculation of the triple integral from the highly oscillating function of a general form is researched.Conclusions. The experiments confirm the obtained theoretical results on the error estimation of the approximation triple integral from the highly oscillating function in a general form by the cubature formula. The prospect of further research is to obtain an estimation of the approximation error on wider classes of functions and to prove that the proposed cubature formula is optimal by the order of accuracy.