Let {φ_n(x), n = 1, 2,... } be an arbitrary complete orthonormal system (ONS) on the interval I := [0,1) that consists of a.e. bounded functions. Then there exists a rearrangement {φ_(σ1(n)), n = 1,2,...} of the system {φ_n(x), n = 1,2,...} that has the following property: for arbitrary nonnegative, continuous and nondecreasing on [0, ∞) function ø(u) such that uø(u) is a convex function on [0, ∞) and ø(u) = ο(ln u), u → ∞, there exists a function f ∈ L(I ~2) such that f_(I~2) |f(x,y)| ø(|f(x,y)|) dx dy < ∞ and the sequence of the square partial sums of the Fourier series of f with respect to the double system {φ_(σ1(m))(x)φ_(σ1(n))(y), m,n ∈ N} on I~2 is essentially unbounded in measure on I~2.
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