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We obtain the decay bounds for Chebyshev series coefficients of functionswith finite Vitali variation on the unit square. A generalization of the wellknown identity, which relates exact and approximated coefficients, obtainedusing the quadrature formula, is derived. Finally, an asymptotic$L^1$-approximation error of finite partial sum for functions of boundedvariation in sense of Vitali as well as Hardy-Krause, on the unit square isdeduced.

Decay estimate of bivariate Chebyshev coefficients for functions with limited smoothness