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.
翻译:高斯二次曲线的概念可以概括为具有复杂时刻的近似线性功能。 在现有文献之后, 本调查将重新审视这种概括性。 众所周知, 正确定线性函数的( 古典) 高斯二次曲线与正直线函数有关, 也与( 赫米蒂安) 兰佐斯 算法有关。 模拟而言, 线性函数的高斯二次曲线与正式的正向多向多向函数有关, 与非赫米蒂安 兰乔斯 算法与外观战略有关; 此外, 它与最小的部分实现问题有关。 我们将审查这些关联性, 指出在相关背景下独立建立的若干结果之间的关系。 Mismatch Theorem 和匹配状态属性的原始证据是通过直线功能使用正式或正向多向多向函数和高向四方曲线的属性提供的。