Geometric Aspects of Multiple Fourier Series Convergence on the System of Correctly Counted Sets

Abstract

For multiple Fourier series the convergence of partial sums essentially depends on the type of integer sets, to which the sequence numbers of their terms belong. The problem on the general form of such sets is studying in $u$-convergence theory ($u(K)$ - convergence) for multiple Fourier series. An alternative method of summation is based on the concept of the so-called correctly denumarable sets. In the paper some results describing the $u$-convergence relations and convergence on the system or correctly denumarable sets are presented. It is shown that the system of $U(K)$-sets containing a sphere of infinitely increasing radius for fixed $K$ is correctly denumarable. It is established that for the functions satisfying the Lipschitz condition and having a certain growthing $p-k$-variation, the coefficients of multiple Fourier series decrease at the average on the system of $U(K)$-sets faster than it is predicted by their ordinary estimations. It is shown the accurate estimation of the Fourier coefficients of functions of several variables is achieved at a very ``poor'' set of elements of the integer lattice.