Olevs'ka Y. About the spectra of integral operators in spaces of functions of several variables and their application

Object - sequence of singular numbers of integral Fredholm operator, acting in spaces of functions of several variables, coefficients of the Fourier and Fourier-Walsh series of functions of several variables. Purpose - to investigate the behavior of the singular numbers of compact Fredholm integral operators acting in Hilbert spaces of functions of several variables, depending on the properties of the nuclei that generate these operators (in particular, the smoothness and variation nuclei), to determine the rate of convergence to zero of the Fourier coefficients of periodic functions of several variables and the Fourier-Walsh functions. To investigate, in particular, the behavior of the Fourier coefficients with indices belonging to a multi-dimensional lattice with integer components, to establish a connection between the smoothness of the function of many variables and sets of numbers of Fourier coefficients for which these factors have a minimum rate of decrease. In the work are used the methods of the theory of operators in Hilbert spaces, Fourier series theory, analytic number theory. The thesis to be defended is devoted to the upper evaluation of singular values of Fredholm integral operators and Fourier and Fourier-Walsh coefficients of functions also. The -variation is intended for functions of many variables. There has been established the connection between decreasing velocity of integral operator's singular values to zero with features of its kernel, theorems have been proved on function's Fourier and Fourier-Walsh coefficients decreasing velocity. An asymptotic density has been investigated for Fourier coefficients numbers sets, for which minimal order of decreasing is reached. The work is theoretical in nature. The results obtained can be used to further develop the theory of linear Fredholm integral operators. The introduction of a new concept of p-to-variation may have application in studies of various aspects of the theory of functions. New estimates of the rate of decrease of Fourier coefficients are useful in problems where the methods used and the results of harmonic analysis are used.