Numerical Integration of Rapidly Oscillating Functions Using Reconstruction Operators Based on Data On Lines

• Datum: 2025
• Hauptverfasser: Хурдей, Євгенія, Нефьодова, Інна, Заборний, Антон, Летута, Андрій
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Online Zugang: http://mcm-math.kpnu.edu.ua/article/view/347711
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

In modern mathematical modelling of physical and technical processes, the problem of processing and analysing functions of many variables, whose values are known on line systems, is relevant. Digital image processing tasks, in particular the numerical integration of oscillating functions, are no exception. In digital image processing tasks, a significant part of the information about the object under study comes in the form of measurements along individual directions or lines, which is characteristic of tomographic methods, remote sensing and visualization systems. The development and application of effective methods for the numerical integration of rapidly oscillating functions based on data on a line system is an important prerequisite for improving the accuracy of reconstruction, filtering, and analysis of digital images. The research in the article is devoted to the numerical integration of rapidly oscillating functions of several variables. The article presents a cubature formula for the approximate calculation of double integrals of an oscillating exponential function. The cubature formula uses traces on mutually perpendicular lines as function data in its construction. Error estimates are presented for a class of differentiable functions. Much attention is paid to testing the cubature formula for approximate calculation of double integrals of the osclilating exponential function. The results obtained allow us to explain the choice of parameters and confirm the theoretical error estimates. The numerical experiment is of particular importance because it is the basis for analysing and predicting the behaviour of the method in the three-dimensional case. This is due to the fact that with an increase in dimension, the amount of information required about the integrand increases, the error structure becomes more complex, and the computational costs increase significantly. The paper presents a cubature formula for the approximate calculation of triple integrals of an oscillating exponential. The values of the function are given as traces of the function on a system of mutually perpendicular lines. An estimate of the approximation error on a class of differentiable functions is obtained