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) ove r 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 t he rate of this approximation in terms of the oscillation of the function p si(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 ob tain a conjugate version of the well-known Dini-Lipschitz test on uniform c onvergence. We also give estimates in the case where the function f(x, y) i s 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 Academi c Press.