| Abstract: | For classes of functions with convergent Fourier series, the problem of estimating the rate of convergence has always been of interest. The classical theorem of Dirichlet and Jordan for functions of bounded variation assures the convergence of their Fourier series, but gives no estimate of the rate of convergence. Such an estimate was first provided by Bojani
. Here we consider this problem in the case of functions of two variables that are of bounded variation in the sense of Hardy and Krause. The Dirichlet-Jordan test was first extended by Hardy from single to double Fourier series. Now, we provide a quantitative version of it. We prove our estimate in a greater generality, by introducing the so-called rectangular oscillation of a function of two variables over a rectangle. |
