Efficient by Precision Algorithms for Approximating Functions from Some Classes by Fourier Series

Introduction. The problem of approximation can be considered as the basis of computational methods, namely, the approximation of individual functions or classes of functions by functions that are in some sense simpler than the functions being approximated. Most often, the role of an approximant is played by a set of algebraic polynomials or (in the case of a periodic function) a set of trigonometric polynomials of a given order. The ideas and methods of approximation theory are used in various fields of science, especially applied areas, since tasks related to the need to replace one object with another, close in one sense or another to the first, but easier to study, arise very often. The purpose of the paper is consider the problems of approximation of a function, which is given by its values in a certain set of nodal points on a certain interval and belongs to a certain class of functions by trigonometric Fourier series, using the quadrature formulas for calculating integrals of fast oscillating functions on this class of functions, which are optimal in accuracy and close to them. The main attention is paid to the study of the sources of error of the proposed approach to function approximation. Results. Effective approximation algorithms from classes of differentiable functions with the help of Fourier series are proposed, using the Fourier coefficients optimal in accuracy and close to them on the given classes of quadrature formulas for calculating integrals of fast-oscillating functions to determine the Fourier coefficients. The error estimates of the proposed approximation algorithms using the quadrature formulas for calculating the Fourier coefficients of the optimal accuracy and close to them for calculating integrals of fast-oscillating functions from classes of differential functions with given values at the nodes of a fixed grid are presented. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of these classes are given. Conclusions. Efficient by precision algorithms for approximating functions from classes of differentiable functions by means of Fourier series are constructed using the optimal accuracy and close to them quadrature formulas for calculating integrals of fast-oscillating functions from the above classes of functions to calculate the Fourier coefficients. A comprehensive analysis of the quality of the constructed algorithms for approximating functions by finite sums of the Fourier series is carried out. Keywords: function approximation, Fourier series, Fourier series coefficients, quadrature formulas, approximation error.