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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.

Determining the stress state in a circular ring has been a classical topic in the stress analysis literature. Based on the principle of superposition, the results may be obtained by adding known solutions to an associated ring problem, where the boundary stresses on the inner and outer walls of the ring are represented in Fourier series. In this work, the coefficients of the Fourier series are generated through an algorithm based on the fast Fourier transform (FFT). In the case of concentrated loading, the required additional fundamental solutions are derived in closed-form. The presented numerical method allows for accurate and efficient computations of the stress distributions in a circular ring in static equilibrium under arbitrary in-plane loading; and generally, the FFT-based algorithm provides a convenient and versatile tool for handing some two-dimensional problems involving circular boundaries. [ABSTRACT FROM AUTHOR]

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained.

Multiple‐input multiple‐output (MIMO) radars may outperform other radar systems such as phased array radars, in terms of higher resolution, better detection probability in the presence of interferences, better parameter identifiability and more flexibility in beampattern design. Waveform covariance matrix design, because of its role in the beampattern synthesis process, is one of the most important problems in MIMO radar systems. In this study, the authors have proposed a closed‐form solution based on Fourier series coefficients to design a covariance matrix. The resulting covariance matrix fulfils the practical constraints, i.e. positive semi‐definiteness and the uniform elemental power constraint. It also provides performance similar to that of iterative methods, while requires lower computation time and provides better mean square error with respect to other existing closed‐form methods. Eigenvalue decomposition is also utilised to convert the possible resulted pseudo‐covariance matrices (pseudo‐CM), which are not guaranteed to be positive semidefinite, into a covariance matrix. Simulation results show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

The windings of the power transformers are prone to deformation or displacement as a result of huge electromagnetic forces. Regular monitoring of transformers for detecting minor winding movements is crucial for preventing them from developing into major faults and eventually, causing an outage. In this study, a novel procedure is proposed to determine the location and severity of the disk space variation, a frequently occurring mechanical damage in transformers, by developing a two‐level method. At the first level, an accurate model of the winding is determined from sweep frequency response analysis measured data corresponding to the healthy status. Analysis and interpretation of the obtained results by the proposed fault detector algorithm lead to locating the position and severity of the fault, in the next level. The process of determining the winding model from the frequency response analysis data is achieved by a multistage method utilising an enhanced version of magnetic optimisation algorithm, which comprises of utilising Fourier series coefficients and phase functions, and fast calculation of coupled network matrices. Experimental results of two 64 kV transformer windings verify the effectiveness of the proposed method in detecting the winding faults. [ABSTRACT FROM AUTHOR]