Abstract :
Let {ϕk (x), k = 1, 2, . . .} be an arbitrary orthonormal system on [0, 1] that is uniformly bounded
by a constant M. Let T be a subset of [0, 1]2 such that the Fourier series of all Lebesgue integrable
functions on [0, 1]2 with respect to the product system {ϕk (x)ϕl(y), k, l = 1, 2, . . .} converge in
measure by squares on T. The following problem is studied. How large may the measure of T be?
A theorem is proved that implies that for each such system, there is
μ2T 1 −M
−4
(for the d-fold product systems, μdT 1−M−2d, d 2). This estimate is sharp in the class of all
such product systems.
