Method of Calculating Fourier Coefficients of Three Variable Functions Using Tomogram

A method for calculating Fourier coefficients of functions of three variables using tomograms is proposed and investigated. Earlier Lytvyn O.M. proposed and introduced the method of calculating Fourier coefficients of the function of two variables using integrals of this function along a given system of direct. It is believed that these integrals - projections or projection data arrive from a computer tomograph. The theoretical positions of the method devoted to the calculation of the Fourier coefficients of the function of two variables using projective data from the approximate function are given in the paper. A characteristic feature of the proposed method is that it is not based on the inverse transformation of Radon, which is used in computer tomography and does not need to go to polar coordinates when calculating Fourier coefficients. In this paper, for the first time, a generalization of this method is proposed in the case of calculating Fourier coefficients of functions of three variables using tomograms placed on a system of planes parallel to a given system of planes. It is believed that these tomograms come from a computer tomograph. In the three-dimensional case studied in this paper, known integrals of the approximate function are considered in two-dimensional regions lying in planes parallel to the given base planes, which are completely determined by indexes of Fourier coefficients. The explicit formulas for the transition from the Cartesian coordinate system to another Cartesian coordinate system are given in the work, one of whose variables is determined by the indicated base plane, and the other two variables create a two-dimensional Cartesian system located on the specified base plane. To calculate the Fourier coefficients of the function of three variables for different base planes, we obtain two-dimensional integrals. The choice of the necessary base planes is proposed by replacing the rapidly expanding complex exponential factor with periodic piecewise - stable splines of the best approximation.