  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. Finally, an asymptotic
$L^1$-approximation error of finite partial sum for functions of bounded
variation in sense of Vitali as well as Hardy-Krause, on the unit square is
deduced.
