Solution of the Problem of Generalized Localization for Spherical Partial Sums of Multiple Fourier Series

It is well known that Luzin’s conjecture has a positive solution in one dimensional case and it is still open in multidimensional case for the spherical partial sums: \(S_\lambda f(x)=\sum _{|n|^2< \lambda }f_n\,e^{inx},\) where \(f_n\) are the Fourier coefficients of f. Historically, progress with solving the Luzin’s conjecture has been made by considering easier problems. One of such easier problems for \(S_\lambda f(x)\) was suggested by V. A. Il’in in 1968 and this problem is called the generalized localization principle for the spherical partial sums. In 1976, well-known specialists in this field raised the question of the validity of the principle of generalized localization at least for double trigonometric series. In this paper, we give a solution of this problem and indicate a sketch of the proof. In addition, a result is given on the convergence of spherical partial sums for functions from the Sobolev class.