This research and survey work deals exclusively with the study of the approximation of generalized multivariate Poisson–Cauchy type singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. These operators are not in general positive linear operators. In particular we study the rate of convergence of these integral operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these integral operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of approximation. The above properties are studied with respect to \(L_{p}\) norm, \(1\le p\le \infty \).

这项研究和调查工作专门研究广义多元Poisson–Cauchy型奇异积分对单位单元算符的逼近。在这里，我们定量研究了它们的大多数近似性质。这些算子通常不是正线性算子。特别是，我们研究了这些积分算子对单位算子的收敛速度，以及相关的同时逼近。这些是通过杰克逊型不等式以及所涉及函数的高阶偏导数的平滑度的多元高阶模量给出的。我们还研究了这些积分算子的全局光滑度保持性质。这些多元不平等几乎是尖锐的，在一种情况下，这种不平等是尖锐的。此外，我们给出了近似误差的Voronovskaya型的渐近展开。研究以上特性\（L_ {p} \）规范\（1 \ le p \ le \ infty \）。