Decay estimate of bivariate Chebyshev coefficients for functions with limited smoothness

We obtain the decay bounds for Chebyshev series coefficients of functions with finite Vitali variation on the unit square. A generalization of the well known identity, which relates exact and approximated coefficients, obtained using the quadrature formula, is derived. An asymptotic $L^1$-approximation error of finite partial sum for functions defined on the unit square is deduced.