The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realization problem. We will review these connections pointing out the relationships between several results established independently in related contexts. Original proofs of the Mismatch Theorem and of the Matching Moment Property are given by using the properties of formal orthogonal polynomials and the Gauss quadrature for linear functionals.

高斯正交的概念可以推广为具有复杂矩的近似线性函数。根据现有文献，本次调查将重新审视这种概括。众所周知，正定线性泛函的（经典）高斯求积与正交多项式以及（埃尔米特）Lanczos算法有关。类似地，线性泛函的高斯正交与形式正交多项式以及具有超前策略的非Hermitian Lanczos算法相关；而且，它与最小的部分实现问题有关。我们将回顾这些联系，指出在相关上下文中独立建立的多个结果之间的关系。