Multiple Fourier series and Fourier integrals with non-separable variables

Background. Integral transforms for functions of several variables are an actively developing area of mathematical analysis. Numerous applications in integral transforms method for solving equations of mathematical physics, in signal processing of engineering require improvement of the theoretical apparatus of integral transforms. The article proposes to depart from the concept of symmetric integral transforms, the inverse transform of which has a Hermitian conjugate kernel to the kernel of the direct transform. In this paper, we find a decomposition of a function of two variables for the Fourier integral with a grouping of harmonics by spectra on concentric circles. A similar idea is implemented in the article for the theory of multiple Fourier series, when the harmonic frequencies are grouped along the border of a square or rhombus.
Materials and methods. The paper presents the proof of integral transforms with non-separable variables based on the decomposition theorem. In this case, the calculation of integrals with respect to spectral parameters is carried out by carried out to the polar coordinate system or to its generalization. In this case, the integral of (n - 1)-dimension can be calculated analytically and it leaves one integral on the polar axis in the reversal formula.
Results. Formulas for inversion of the multiple Fourier integral are constructed. Their peculiarity is that integration is carried out along the polar axis, whereas in the classical inversion formula integration is carried out along n-dimensions manifold. Similarly, in the theory of multiple Fourier series, a series expansion is obtained in harmonics are grouped in a certain way and then summed in a closed form. The article suggests various ways of grouping the harmonic components of a multiple Fourier series, which allowed us to obtain new formulas for inversion.
Conclusions. New inversion formulas for the multiple Fourier series and the multiple Fourier integrals are proved. These formulas can be used in the obtaining of discrete analogs of multiple Fourier integrals for the purpose of their application in the processing of 2D- and 3D-signals.