Approximation by rectangular partial sums of double conjugate Fourier series (Q1971650)

The author examines the degree of approximation of the conjugate function \(\widehat f^{10}(x,y)\) by the symmetric rectangular partial sums of the series conjugate to double Fourier series in terms of oscillation of the function \[ \psi^{10}_{xy}(f, u,v)= f(x- u,y- v)- f(x+ u, y-v)+ f(x- u,y+ v)- f(x+ u,y+ v) \] over appropriate rectangles of the torus \(\mathbb{T}^2\), where \(f\in L^1(\mathbb{T}^2)\) is bounded. In particular, he obtains a conjugate version of the Dini-Lipschitz test on uniform convergence. He also obtains estimates in the case where the function \(f(x,y)\) is of bounded variation in the sense of Hardy and Krause.