Algorithms for approximating a function from inaccurate observations

Abstract

This paper is devoted to the approximation of a function by a trigonometric polynomial based on its imprecise values at specially selected points. To solve this problem, methods for measuring the parameters of differential methods using inaccurate observations in special cases of selected points, previously proposed by the authors in their works, were using. Two ends of the approximation function defined on a segment are considered. The first method is to infer the distribution of m points on the segment where the function is given. The second method is to study a large number of numbers of observations at m points dividing a segment into a finite, identical number of parts. For these approximations of constructions, the root-mean-square deviations of the function from trigonometric polynomials are estimating and the rate of their convergence as m tends to infinity is estimating as O(m–1/2). It is shown that these observations in both cases have the form of a number of observations and the computational characteristics differ significantly. So, in the first case, this is equal to O(m3/2), and in the second case — O(m2 ). On the other hand, the use of the fast Fourier transform allows in the second case to significantly reduce the computational complexity. Thus, the task of approximating a function from inaccurate observations of their values at selected points is multicriteria. In addition, its solution depends on the method of selecting observation points, i.e. from the experimental planning procedure. In the second case, it is basing on a large number of observations in small vicinities of the points of division of the segment into a finite number of equal parts.