Наближення розривних функцій розривними сплайнами методом мінімакса

Abstract

Розроблено метод наближення функцій однієї змінної, що мають розриви першого роду, за допомогою розривних лінійних апроксимаційних сплайнів. В якості експериментальних даних виступають односторонні границі заданих вузлів. Пропонується шукати такі параметри розривного сплайна, щоб наближення було найкращим у тому чи іншому сенсі. Для розв’язування цієї задачі в даній роботі використовується методом мінімакса. Детально описані чисельні експерименти, які підтверджують ефективність запропонованого методу.The problem of one variable function approximation having discontinuities of the first kind is solved in the article. The experimental data are the unilateral boundaries of the given nodes. The idea is that the functions of certain classes are better approximated by functions belonging to these classes too. Those discontinuous functions with discontinuities of the first kind are naturally approximated by discontinuous splines, and not by sums of infinitely differentiable functions, as is done in the method of approximation by Fourier sums. This allows you to get rid of the phenomenon of Gibbs. Therefore, the method proposed in the paper makes essential use of discontinuous linear approximation splines. The discontinuous constructions proposed in this paper for approximation of discontinuous functions have the same order of smoothness as the object under study. It is proposed to search for such parameters of a discontinuous spline, so that approximation is best in one or another sense. To solve this problem, we use the minimax method in this paper. Numerical experiments that confirm the effectiveness of the proposed method are described in detail. In further studies, it is planned to develop methods for identifying break points using the minimax method and to develop a theory of approximation of discontinuous functions of two variables and algorithms for identifying lines of discontinuity. The developed methods can be used to solve problems using remote methods. For example, in flaw detection in the detection of cracks in industrial products using nondestructive counter, in many problems of geophysics in establishing the location of boundaries that separate blocks with different physical properties that characterize the internal structure of the Earth. In medical computed tomography, when studying the internal structure of the body, it is useful to use its heterogeneity, i.e. different density in different parts of the body.