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.
翻译:我们获得了Chebyshev系列函数的衰变界限,在单位方形上有有限的维塔利变异系数。我们推断出使用二次方形公式获得的与精确和近似系数有关的众所周知的身份。最后,在单位方形的维塔利和Hardy-Krause意义上的受约束变异功能的有限部分和部分金额的零用1美元(相当于1美元)的零用量误差。