We extend the close interplay between continued fractions, orthogonal polynomials, and Gaussian quadrature rules to several variables in a special but natural setting which we characterize in terms of moment sequences. The crucial condition for the characterization is the commutativity of the multiplication operators on finite polynomial subspaces modulo an ideal. Moreover, starting from the orthogonal polynomials or the three-term recurrence, our method constructs a sequence of moment sequences that provides an approximation of the maximal order for recovering the moment sequence that defines the orthogonality.

翻译：在一种特殊但自然的设定下，我们将在矩序列的表征中扩展连分数、正交多项式与高斯求积规则之间的紧密联系，其中关键条件是有限多项式子空间上的乘法算子模一个理想具有交换性。此外，从正交多项式或三项递推关系出发，我们的方法构建了一系列矩序列，该序列提供了恢复定义正交性的矩序列的最大阶近似。