# The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization

## TL;DR

This paper reviews the generalization of Gauss quadrature to complex linear functionals, exploring its connections with formal orthogonal polynomials, non-Hermitian Lanczos algorithms, and minimal partial realization, providing new proofs of key theorems.

## Contribution

It offers a comprehensive survey of Gauss quadrature for linear functionals, highlighting its links with various mathematical concepts and presenting original proofs of important theorems.

## Key findings

- Connections between Gauss quadrature and formal orthogonal polynomials clarified.
- Relationship with non-Hermitian Lanczos algorithm and minimal partial realization established.
- Original proofs of the Mismatch Theorem and Matching Moment Property provided.

## Abstract

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.