Approximation by rectangular partial sums of double conjugate Fourier series

We consider functions f(x, y) bounded and measurable on the two-dimensional torus T-2. The conjugate function (f) over tilde(10)(x, y) with respect to the first variable is approximated by the rectangular partial sums (S) over tilde(mn)(10)(f; x, y) of the corresponding conjugate series as m, n tend to proportional to independently of one another. Our goal is to estimate the rate of this approximation in terms of the oscillation of the function psi(xy)(10)(f; u, v):= f(x - u, y - v) - f(x + u, y- v) + f(x - u, y + r) - f(x + u, y + v) over appropriate subrectangles of T-2. in particular, we obtain a conjugate version of the well-known Dini-Lipschitz test on uniform convergence. We also give estimates in the case where the function f(x, y) is of bounded variation in the sense of Hardy and Krause. Results of similar nature on the one-dimensional torus T were proved in [7]. (C) 2000 Academic Press.