Approximation by Rectangular Partial Sums of Double Conjugate Fourier Series

Abstract

We consider functions f(x, y) bounded and measurable on the two-dimensional torus T2. The conjugate function f10(x, y) with respect to the first variable is approximated by the rectangular partial sums s10mn(f; x, y) of the corresponding conjugate series as m, n tend to ∞ independently of one another. Our goal is to estimate the rate of this approximation in terms of the oscillation of the function ψ10xy(f; u, v)≔f(x−u, y−v)−f(x+u, y−v)+f(x− u, y+v)−f(x+u, y+v) over appropriate subrectangles of T2. 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].