On Divergence to Infinity of Subsequences of Partial Sums of Orthogonal Fourier Series

Let \φ ⱼ(x,y) =1∞ { φ j ( x , y ) } j = 1 ∞ be a uniformly bounded orthonormal system on [0,1]² [ 0 , 1 ] 2 , ₙ\ₙ₌₁∞ { A n } n = 1 ∞ be sequence of bounded subsets of Z₊=\1, 2, \ Z + = { 1 , 2 , ⋯ } and ₙ\ { N n } denote the number of elements in Aₙ A n . Let also LAₙ(x,y) L A n ( x , y ) and S_Aₙ(f; x,y) S A n ( f ; x , y ) denote, respectively, the Lebesgue function and the partial sum of the Fourier series of a function f with respect to the system \φ ⱼ(x,y)\ { φ j ( x , y ) } corresponding to the indices from Aₙ A n