We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions ${\tilde{\phi }}_j$ and trigonometric polynomials ${\phi }_j$. The class of such operators includes classical interpolation polynomials (${\tilde{\phi }}_j$ is the Dirac delta function), Kantorovich-type operators (${\tilde{\phi }}_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on ${\tilde{\phi }}_j$ and ${\phi }_j$, we obtain upper and lower bound estimates for the $L_p$-error of approximation by quasi-interpolation operators in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, $K$-functionals, and other terms.

我们研究了由分布/函数生成的一般多元周期性准插值算子的近似性质 ${\stackrel{～}{\phi }}_j$ 和三角多项式 ${\phi }_j$. 此类运算符包括经典插值多项式 (${\stackrel{～}{\phi }}_j$ 是狄拉克 delta 函数），Kantorovich 型运算符（${\stackrel{～}{\phi }}_j$是一个特征函数）、与小波构造相关的缩放扩展等。在不同的兼容条件下${\stackrel{～}{\phi }}_j$ 和 ${\phi }_j$，我们获得了上限和下限估计 ${升}_{磷}$- 准插值算子在最佳和最佳单边近似、经典和分数平滑模量方面的近似误差， $钾$-泛函和其他术语。