Gibbs' phenomenon and surface area

If a function $f$ is of bounded variation on $T^N (N\geq 1)$ and $\{{\varphi}_n\}$ is a positive approximate identity, we prove that the area of the graph of $f*{\varphi}_n$ converges from below to the relaxed area of the graph of $f$. Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities.