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Our aim in this paper is to extend a known theorem about the convergence of subsequences of the partial sums of the Fourier series in one variable of class L 2 to Fourier series in two variables of the same class, (1, p. 396). The theorem asserts that for each function ƒ in L 2, there is a sequence {m V } of positive integers of upper density one such that
Smv(X;ƒ)
converges to ƒ almost everywhere where sm(x;f) denotes the mth partial sum of the Fourier series of ƒ.