Polynomials Orthogonal on the Unit Circle in Numerical Analysis & Signal Processing

数值分析在越来越多的领域发挥着重要作用，部分原因是计算在数学科学中的作用不断扩大。数值分析中的许多算法使用在实区间(其子集)上正交的多项式。这些算法包括多项式最小二乘逼近、高斯求积、Lanczos和共轭梯度法。它计划开发类似的算法，使用在单位圆上正交的多项式，也称为Szego多项式。Szego多项式与三角多项式的逼近有关，与区间上正交的多项式与(代数)多项式的逼近类似。Szego多项式满足一个少项递推关系。这使得它们在计算中使用起来很有吸引力。它们已经成功地应用于计算酉阵的特征值的算法中，这是信号处理中某些频率估计方法所出现的问题。

Numerical analysis is of importance in an increasing number of areas, owing in part to the ever - expanding role of computation in the mathematical sciences. Many algorithms in numerical analysis use polynomials orthogonal on (a subset of) a real interval. These algorithms include schemes for polynomial least squares approximation, Gaussian quadrature, and the Lanczos and conjugate gradient methods. It is planned to develop analogous algorithms that use polynomials orthogonal on the unit circle, also known as Szego polynomials. Szego polynomials are related to approximation by trigonometric polynomials, similarly as polynomials orthogonal on an interval are related to approximation by (algebraic) polynomials. Szego polynomials satisfy a recurrence relation with few terms. This makes them attractive to use in computations. They have already been applied successfully in algorithms for computing eigenvalues of unitary matrices, a problem that arises in certain frequency estimation methods in signal processing.